物理学经典书籍 Discrete Quantum Electrodynamics - C.Francis

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Discrete Quantum Electrodynamics

Charles Francis

Abstract:The purpose of this paper is to construct a quantum field theory suitable for describing quantum electro-

dynamics and Yang-Mills theory in a form which satisfies the conditions of the Millennium prize offered by

the Clay Mathematics Institute as described by Jaffe and Witten [12] , by showing that it satisfies ‘ axioms at

least as strong as those cited by ’ Wightman [18] and by Osterwalder and Schrader [14] , and by observing

that this form of field theory has no mass gap. The definitions provide a model for relativistic quantum

mechanics which supports a form of relativistic quantum field theory, but which does not depend on the sec-ond quantisation of a “matter wave”. Continuous laws of wave mechanics are found in model of discrete

particle interactions which does not involve waves, or the quantisation of interacting fields. Newton’s first

law and conservation of momentum are established from the principle of homogeneity. Maxwell’s equations

are derived from the simple interaction in which a Dirac particle emits or absorbs a photon, showing that

the renormalised mass and coupling constant are equal to their bare values. Feynman rules are calculated

for the discrete theory and give the predictions of the standard renormalised theory. Quark confining inter-

actions are described for qed and for an adaptation of Yang-Mills theory.

Charles Francis

Clef Digital Systems Ltd.

Lluest, NeuaddlwydLampeter

Ceredigion

SA48 7RG

20/8/01



Discrete Quantum Electrodynamics

1 BackgroundIt has long been known that there is no finite unitary representation of the Lorentz group, and this paper

seeks to resolve the problems in constructing field operators on covariant Hilbert space by defining equiv-

alence classes of operators on an infinite family of finite dimensional Hilbert spaces. Lorentz transformationwill invoke a different Hilbert space on which the transformed field operator is unambiguously defined. Alattice is used to define the basis for each Hilbert space, but there are important differences between discrete

quantum field theory as described here and, e.g. lattice quantum field theory as developed by Wilson and

others and described by e.g. Montvay & Münster [13] . The physical structure to which this mathematical

model relates is also described using an “information space” interpretation in which reality exists but quan-

tum mechanics describes the information we have about it. The reader should distinguish interpretation from

the mathematical construction based on the given definitions which is the subject of this paper, but interpre-

tation is necessary because the mathematical structure applies to each observer’s information space while itcould not apply to ontological space-time, and to avoid confusion with interpretational remarks found in

other treatments which cannot be applied here. The interpretation is orthodox, being based on Heisenberg’sdiscussion of the Copenhagen interpretation in [9] and the Dirac-Von Neumann approach to it [10][2] . It

does not use Bohr’s complementary, Jordan’s second quantisation of ‘matter fields’, nor have a dependency

on a classical Lagrangian or action principle. Commutation relations are found from the definition of momentum states as linear combinations of position states, not simply imposed by canonical quantisation.

As distinct from lattice qed, discrete qed uses Minkowsky rather than Euclidean co-ordinates, and uses a

bounded momentum space with an automatic cut-off. An energy cut-off follows from the mass shell condi-

tion, but there is no cut-off in the off mass shell energy which appears in the perturbation expansion, since

this is an abstract parameter in a contour integral ( section 18 ). A discrete model cannot be manifestly cov-

ariant. Manifest covariance is not necessary since the lattice is not physical and is observer dependent. It isrequired that the laws of physics are the same in all reference frames, and covariance is redefined to allowan invariant hermitian product on a space of covariant functions. Points in one observer’s information space

do not transform to those in another’s and operators for position in different reference frames do not com-mute. Classical vector observables exist on a scale where the lattice appears as a continuum. This scale

appears many orders of magnitude smaller than experimental errors. Manifest covariance and the renormal-

ised formulae of the standard model are recovered on letting lattice size go to infinity and lattice spacing go

to zero. The Landau pole appears in this limit but only after reordering series where we do not have absolute

convergence and does not indicate anything more serious than the breakdown of an iterative solution to a

non-perturbative equation. In fact the Landau pole does not appear if lattice spacing is bounded below by a

fundamental interval of proper time in the interactions between particles, and it is suggested that this is in

fact the case in a subsequent paper [7] , where a minimum proper time between interactions is related to uni-

fication with gravity.Using finite dimensional space the definition of linear field operators is unproblematic, as they are oper-

ators, not operator valued distributions. The field operators constructed here will be used to describe

interactions between particles using the interaction picture. Although formally similar to quantised freefields, they will be used to describe the potential for the creation or annihilation of a particle in an interac-



Discrete Quantum Electrodynamics 2

tion. This cannot be reconciled to interpretational statements which are sometimes made about models

which arise from the quantisation of classical fields, such as “ The free field describes particles which do not

interact ” (Glimm & Jaffe p 100 [8]); it will be appreciated that while the formulae of the present construc-tion are extremely like the formulae which arise in such models and give the same physical predictions in

the appropriate limits, the meaning of these formulae is quite different.

Although apparently innocuous, perhaps the most significant mathematical difference between this andother formulations of field theory is that time is a parameter, as in non-relativistic quantum mechanics, andeach Hilbert space is formulated for synchronous states at time t . The U-matrix is strictly a map

where , so that unitarity does not apply (conservation of probability is

required). Homomorphically identifying introduces what has perhaps been the centralproblem in the construction of quantum field theory in 4 dimensions, namely the indefinability of the equal

point multiplication between the field operators (which is definable but would invalidate the limit as latticespacing goes to zero). This is resolved pragmatically by normal ordering all equal point multiplications of

field operators. Τhis non-linear condition is motivated physically by saying that a particle created at can-

not interact again at the same instant. This has an important effect upon renormalisation and gives physicalmotivation to the method of Epstein & Glaser [17][5] in which is replaced with a con-

tinuous switching function which is zero at , removing loop divergences. The analysis of the origin

of the ultraviolet divergence is, for practical purposes, that given by Scharf [17] , namely the incorrect use

of Wick’s theorem. The difference between this treatment and Scharf is that here the limiting procedure usesa discrete sum whereas Epstein and Glaser use a continuous switching function, and while Scharf says

(p163) “the switching on and off the interaction is unphysical”, here the switching off and on of the interac-tion at is a physical constraint meaning that only one interaction takes place for each particle in

any instant, i.e. that the interaction operator at time t cannot act on the result of itself. In practice this is done

by normal ordering all equal time products of fields. Thus an annihilation operator at time t does not act on

a creation operator at the same time. As with the method of Epstein & Glaser, this leads directly to a pertur-bation expansion in which the terms are finite and similar in form to the standard “renormalised” expansion.

Thus we will state that the correct description of physical processes in qft uses an interaction Hamiltonian

such that . is not a linear operator so we do not have . In other respects H is

much as in ordinary qed. Unitarity does not apply to non-linear operators, and it will be shown that H pre-

serves the norm as required by conservation of probability. Linearity of time evolution is required to avoid

a dependency on histories; the interactions of a particle created yesterday should be no different from thoseof one created the day before. Despite the apparently radical nature of non-linearity it will be seen that the

time evolution proposed here is linear for states created in the past, and only distinguishes the interaction of a particle created “now” from those of particles created “not now”.

It will be clear from the use of families of finite dimensional Hilbert spaces that the present constructiondoes not obey the Wightman axioms, which specify a single covariant Hilbert space. It will be clear once

transformations between observer dependent Hilbert spaces have been specified, and field operators have

been defined on such Hilbert spaces, that the model obeys axioms at least as strong as those cited by Wight-man [18] and by Osterwalder and Schrader [14] . It is immediate that the algebras of operators defined by

15.18 for the photon, and by 16.1 together with 16.5 and 16.8 for the Dirac field obey the Haag-Kastler axi-

oms given in [8] after redefining the Lorentz group as specified in section 8 . It will be seen that bare particle

* t ( )U t 1 t 2,( ):* t 1( ) * t 2( )→ t 1 t 2≠

t ∀( )* t ( ) * =

x0n

θ t 1 t –( ) θ t t 1–( )–

t t 1=

x0n x0

j=

H x( ) H 2 x( ) 0= H H x( ) 0=




masses are a parameter of this model, independent of lattice spacing, and they are not changed by renormal-

isation. The existence of a mass gap is equivalent to the statement that the model has no zero mass particles.

The model allows a zero mass neutrino (even if there is not one in nature), and since photons, and other inter-

mediate vector bosons such as the gluon, are here treated as particles on an equal footing with fermions, it

is obvious that this model has no mass gap. Indeed there will always be eigenvalues of the Hamiltonian con-

taining an indefinite number of photons at an energy threshold below that for particle creation. Naturally if a massive particle is present, the spectrum of the Hamiltonian will always be above that mass.

2 Information SpaceIt is important to distinguish the definitional properties of information space from postulates about the

behaviour of matter. We make no assumption of an ontological background in which matter is placed and

define a reference frame as the set of potential results of measurement of position. In this view geometry is

simply and literally world ( geo- ) measurement (- metry ); to understand geometry we must study how observ-

ers measure space-time co-ordinates. Each observer has a clock, which is, without loss of generality, the

origin of his co-ordinate system and measures proper time for that observer. Each event is given a co-ordi-

nate by measuring the time taken for light to travel to and fro the event. Now suppose there is somefundamental minimum unit of time. Then the measured co-ordinates are integer multiples of that unit and

the co-ordinate system is a lattice consisting of measured events and potential measured events in discretemultiples of that unit, bounded by the time period in which the observer carries out measurements. Thus

each observer’s reference frame is defined as a lattice determined by the finite resolution with which it is

possible to measure time; it is part of information space of a particular observer, not prior ontology. For sim-plicity we will use a cubical lattice, although any lattice can be used as appropriate to the measuring

apparatus and chosen co-ordinate system of an individual observer.

The notion that space-time appears not as an invariant background but as an observer dependent set of potential measurement results, is in strictly accordance with the orthodox interpretation of quantummechanics. In Dirac’s words “ In the general case we cannot speak of an observable having a value for a

particular state, but we can .... speak of the probability of its having a specified value for the state, meaning

the probability of this specified value being obtained when one makes a measurement of the observable”

[3] . When this statement is applied to the position observable, it follows that position exists only in meas-

urement of position, and not as part of background geometry.The measurement of time and position is sufficient for the study of many (it has been suggested all) other

physical quantities; for example a classical measurement of velocity may be reduced to a time trial over a

measured distance, and a typical measurement of momentum of a particle involves plotting its path in a bub-

ble chamber. In practice there is also a bound on the magnitude of the result, and we take the results of

measurement of position to be in a finite region , in units of lattice spacing, χ. Without loss of gen-

erality define

Definition: The coordinate system is for some , where

.

Ν 0 3⊂

Ν ν– ν,( ] ν– ν,( ] ν– ν,( ] 0 3⊂⊗ ⊗= ν 0∈

ν– ν,( ] x 0 ν– x ν≤<∈{ }=




There is no significance in the bound, ν, of a given co-ordinate system Ν. If matter goes outside of N it

is merely moving out of a co-ordinate system, not out of the universe. Generally it is possible to describe its

motion in another co-ordinate system with another origin. Even if we do not intend to take the limit ,

Νmay chosen large enough to say with certainty up to the limit of experimental accuracy, that it contains

any particle under study for the duration of the experiment. We ignore matter outside of Νand impose con-

servation of probability as usual.Definition: Let 0ý be the set of finite discrete coordinate systems of sufficient size to include all particles

under study which can be defined by an observer by means of physical measurement. 0 includes curvilin-

ear as well as rectangular coordinates.

The intention is to compose a manifold out of the collection of such co-ordinate systems. In general thiswill require non-Euclidean metric [7] , but for the purpose of this paper, co-ordinate systems in 0 will be

defined using a flat metric, and will be regarded as defining a tangent space. We will not here study the con-nection between coordinate systems with different origins, but merely note the intention that this model be

compatible with any Lorentzian manifold.

Definition: For any point is the ket corresponding to a measurement of position with result x.is called a position ket.

Definition: Let be the empty ket or the vacuum state, a name for a state of no particles.

In the absence of information, we cannot describe the actual configuration of particles; kets are names or

labels for states, not descriptions of matter. The significance is that the principle of superposition will be

introduced as a definitional truism in a naming system, not as a physical assumption. Although kets are not

states, but merely names for states, we loosely refer to kets as states in keeping with common practice when

no ambiguity arises. In a typical measurement in quantum mechanics we study a particle in near isolation.The suggestion is that there are too few ontological relationships to generate the property of position. Then

position does not exist prior to the measurement, and the measurement itself is responsible for introducinginteractions which generate position. In this case, prior to the measurement, the state of the system is notlabelled by a position ket, and we define labels containing information about other states – namely the infor-

mation about what would happen in a measurement.In scientific measurement we set up many repetitions of a system, and record the frequency of each result.

Probability is interpreted here simply as a prediction of a frequency distribution, so a mathematical model

must associate a probability with each possible result. The probability of a given result can be used to attach

a label to the state in the following way:

Definition: Let

Definition: Construct a vector space, * , over % , with basis * 0. The members of * are called kets.

This is trivial because * 0 is finite. * has dimension .

Definition: , the braket is the hermitian form on * defined by its action * 0

, , 2.1

Definition: The position function of the ket is the mapping defined by

Later the position function will be identified with the restriction of the wave function to 0 , but we use the

ν ∞→

x Ν x| ,∈ x|

|

*0 x| x Ν∈{ } | { }∪=

8 ν3 1+

f | g| *∈,∀ g f |

x y Ν x y | ,∈,∀ δ xy= | 1= x | x | 0= =

f | *∈ Ν % → x Ν∈ x,∀ x f | →




term position function, because it is discrete and because a wave is not assumed. We observe that

P ( x| f ) = 2.2

has the properties of a probability distribution, and we associate the ket with any state such that

the probability that a measurement of position has result x is given by .

In this interpretation the position function (and more generally the scalar product) can be identified with

a proposition in a many valued logic [16] . The value of the position function is the (complex) truth value of the proposition. Classical logic applies to sets of statements about the real world which are definitely true

or definitely false. For example, when we make a statement

2ý ( x ) = The position of a particle is x 2.3

we tend to assume that it is definitely true, or definitely false. If it is actually the case that 2ý ( x) is definitely

true or false, then classical logic and classical mechanics apply. But in quantum mechanics it is not, in gen-eral, possible to say that a particle has a definite position when position is not measured, and we can only

consider sets of propositions to describe hypothetical measurement results

(ý ( x ) = If a measurement of position were to be done the result would be x 2.4

The only empirical states are those which are measured, and only for measured states can we make def-

inite statements with in the form of 2 ( x ) and having truth values from the set . But we can categoriseother states according to the likelyhood of what might happen in measurement by using the structure of

Hilbert space and a probability interpretation based on the inner product. Thus we construct new proposi-tions by identifying the vector sum with weighted logical OR between propositions. We say

that is the truth value of the proposition (ý ( x ). Multiplication by a scalar only has meaning as a

weighting between alternatives and we have that such that , labels the samestate (or set of states) as . Probabilities defined from 2.2 are invariant under local U (1) gauge symmetry

and interactions must preserve this symmetry since it describes redundant information appearing in themathematical model but without physical meaning. The information space interpretation inverts the meas-

urement problem. Collapse just represents a change in information due to a new measurement. But

Schrödinger's equation does require explanation – interference patterns are real. It will be derived in 9.1 . It

is routine to prove that

2.5

2.6

So long as we recognise that these are just rules, a calculational device which says nothing about meta-physics there is no inconsistency, ambiguity or other problem with the use of Hilbert space to categoriseunmeasured states. The wave property of superposition comes from the logic rather than the metaphysic.

We can make statements in the form of ( ( x ) and interpret the probability amplitude for such statements as

a truth value, but we cannot in general make statements in the form 2ý ( x )ï The interpretation begs the ques-

tion what is wrong with 2ý ( x )? Is there no such thing as numerical position except in certain states, or is

there no such thing as a particle? If in fact there is no background continuum then there is no reason to rejectthe notion that the universe is composed of particles.

f x | 2

f f | ---------------------

f | *∈

x Ν∈∀ P x f ( )

0 1,{ }

a f | b g| +

x f | f | *∈∀ λ % ∈∀, λ 0≠ λ f |

f |

g f | g x | x f | x Ν∈∑=

x| x Ν∈∑ x | 1=




3 Momentum SpaceIn this treatment momentum is defined mathematically, not empirically. The correspondence with a

measurable physical quantity is to be made by demonstrating conservation of momentum ( section 14 ).Although the formulae derived below are largely standard it is necessary to show them here to distinguish

definitional properties of a mathematical structure from physical postulates, because canonical quantisation

is not assumed, and because this treatment uses finite dimensional Hilbert space which has useful propertieswhich do not hold in infinite dimensional space. In particular it is worth observing that representations of the Dirac delta function on momentum space are ordinary functions (a distribution theoretic treatment is

recovered by allowing lattice spacing to tend to zero after calculation of probabilities).

Definition: Momentum space is the 3-torus . Elements of momen-

tum space are called momenta.

Definition: For each value of momentum , define a ket , known as a plane wave state, by the

position function

3.1

The unit of momentum is 1/length. If lattice spacing is χ in conventional units, then there is a bound of

on the magnitude of each component of momentum. In general this is a bound on the magnitude of momen-

tum which can be measured with apparatus at any given resolution, but later we will interpret χ as a

fundamental length determined by the interactions of matter. Then χ is a fundamental physical constantwhich is invariant in all reference frames, and is an absolute physical bound on momentum. The impli-

cations for transformations of a bound on momentum with be considered in more detail in section 8 . We willuse units in which . The expansion of in the basis * 0 is calculated by using the resolution of

unity, 2.6

3.2

Definition: For each ket define the momentum space function as the Fourier transform, ,

3.3

By 2.6 , F can be expanded as a trigonometric polynomial

3.4

by 3.1 and 3.3. By 3.4 and 3.1

3.5

Rewriting 3.5 in the notations of 3.3 and 3.1

3.6

Μ π– π,[ ] π– π,[ ] π– π,[ ] 4 3⊂⊗ ⊗=

Μ∈ p|

x p | 1

2π------

=

32---

ei x p⋅

π χ ⁄

π χ ⁄

χ 1= p|

p| x| x p | x Ν∈∑ 1

2π------ 32---

e i x p⋅ x| x Ν∈∑= =

f | F :Μ % →F p( ) p f | =

F p( ) p x | x f | x Ν∈∑ 1

2π------ 3

2---

x f | e i– x p⋅

x Ν∈∑= =

12π------ 3

2---

d 3 p F p( )e i x p⋅

Μ∫ 1

8π3--------- d 3 p x f | e i– y p⋅ e i x p⋅

y Ν∈∑

Μ∫ x f | = =

x f | d 3 p

Μ∫ x p | p f | =




For any integrable there is a unique position function given by

3.7

In general 3.7 is not invertible, but 3.4 picks out a unique invertible member of the equivalence class of func-

tions with the same position function.Definition: Members of this class are called representations of the momentum space wave function. 3.4 isthe analytic momentum space function in a given reference frame.

The cardinality of the plane wave states is greater than the cardinality of * 0, so is not a basis.

A convenient basis in momentum space is

Then it is natural to define another invertible representation of the momentum space wave function:

Definition: The discrete momentum space wave function is defined by the DFT (Discrete Fourier

Transform)

where is such that 3.8

Proof: 3.8 is a representation because 3.7 reduces to

The use of the DFT extends the definitions of Μand Ν in a non-physical manner so that both the coordi-nate space and momentum space wave functions are periodic. This has no physical meaning. Since energy

is conserved in physical measurement ( section 18 ), momentum is bounded by the total energy of a system.The probability of finding a momentum above this bound is zero. In general the analytic representation does

not have support with such a bound, but exhibits Gibb’s ripples. The analytic representation is not

directly related to probability, but the discrete representation which uses a basis in momentum space is aprobability amplitude, and we assume that for physically realisable states the support of the discrete repre-

sentation of has a suitable bound. The importance of this bound is further discussed in section 8 .

Because the model is defined in position space with the hermitian product given by 2.5 all predictions are

identical for each equivalent momentum space wave function. For example, the components of the experi-

mental momentum operator are given by

for ,

Then

3.9

F ′:Μ % →

x f | 12π------ 3

2---

d 3 p F ′ p( )e i x p⋅

Μ∫ =

F ′:Μ % →

p| p Μ∈{ }

* Μ p| p Μ∈ and for i 1 2 3, ,= ν p i

π------- < ∈

=

F D p( ) F p b( )= p b * Μ∈ for i 1 2 3 0 p i p ib– π ν ⁄ <≤, ,=

12π------ 3

2---

d 3 p F D p( )e i x p⋅

Μ∫ 1

2π------ 3

2---

p f | e i x p⋅

p| *∈ Μ∑=

12π------ 3

x′ ′′ ′ f | e i x ′ p⋅– e i x p⋅

x ′ Ν∈∑

p| *∈ Μ∑=

x f | =

p f |

p f |

i 1 2 3, ,= P ii–

2---- x| x 1 i+ | x 1 i– |–[ ]

x Ν∈∑=

P i p| i–2---- x| x 1 i+ | x 1 i– |–[ ] p|

x Ν∈∑ x| x p | p isin

x Ν∈∑ p isin p| = = =




So the eigenvalue of momentum is for p much less than . There is some indication that mag-

nitude of the discrete unit of time for a particular particle is twice the Schwarzschild radius , where

m is the mass of the particle and G is the gravitation constant [7] . Then an electron with a difference of 0.1%between p and would have an energy of eV, which may be thought

unrealistic.

By 2.6 and by 3.6

3.10

3.10 is true for all and , so we can define a second form of the resolution of unity

3.11

It follows immediately that

So has the effect of a Dirac delta function on the test space of momentum space functions.Definition: The Dirac delta function is

Explicitly, calculating directly from 3.2

3.12

4 Multiparticle StatesThe rigorous construction of field operators requires the formal definition of multiparticle states and of

creation and annihilation operators. In discrete qed these are operators not operator valued distributions, and

we describe their definition and properties in some detail. Commutation relations will be demonstrated from

the definitions, not imposed canonically. We will define notation in which bras and kets are operators as wellas states. Just as Dirac notation is simpler and more elegant than wave functions, we will see that ket nota-

tion for operators is very powerful and leads to an elegant clarification of field theory.

Definition: For (the external direct product) is the space of kets for states of

particles of the same type.Since contains states of an indefinite number of particles A one particle state cannot be a no

particle state, so by the definition of the braket as a measure of uncertainty

4.1

It follows that the inner product between states of different numbers of particles is always zero. For

, the braket is given by

4.2

p p≈sin π χ ⁄ 4Gm c 3 ⁄

p( )sin 0.055 π χ ⁄ 1.38 10 52×=

f | g| *∈,∀

g f | g x | x 0∈∑ x f | d 3 p

Μ∫ g x |

x 0∈∑ x p | p f | d 3 p

Μ∫ g p | p f | = = =

f | g|

d 3 p

Μ∫ p| p | 1=

q f | d 3 p

Μ∫ q p | p f | =

p q |

δ:7 3 % δ p q–( )→ q p | =

q p |

δ p q–( ) 18π3--------- e i x p q–( )⋅

x Ν∈∑=

n 0 n 0≥,∈ * n*

n⊗=

| *0∈ * n

∀ f | *∈ f | 0=

f | f 1| … f n| , ,( )= g| g 1| … g n| , ,( ) * n∈=

f g | f 1| … f n| , , g1| … gn| , , | f i g i | i 1=

n

∏= =




as is required by the probability interpretation, 2.2 , if each of the particles is independent. Let be

larger than the number of particles in the universe. The space of all particles of a given type γ is The

statement that we can take a value of N greater than any given value is the definition of an infinite sequence,

so, in effect the space of all particles of the same type is Let γ be an index running over each type of

particle. The space of states of particles of type γ is

4.3

Until the treatment of interactions particles of different types γ will be ignored.

5 Creation OperatorsThe creation of a particle in an interaction is described by the action of a creation operator. Creation oper-

ators incorporate the idea that particles of the same type are identical, so that when a particle is created it is

impossible to distinguish it from any existing particle of the same type. There is some advantage in using

creation and annihilation operators to (anti)symmetrise states and at the same time to generate physicalspace, since this ties in with the idea that physical states are created in interactions which are themselvesdescribed as combinations of creation and annihilation operators, and because this will simplify the treat-

ment of quark parastatistics in section 19 . Creation operators are defined by their action on the basis states

. The definition removes arbitrary phase and normalises the two particle state to coincide with 4.2 .

Definition: the creation operator is defined by

5.1

where is to be determined.

Definition: The bra corresponding to is designated by .A bra is here simply an alternate notation for a ket. It will be redefined as an operator in the next section.

Now, by 4.2 and 5.1

5.2

The order in which particles are created can make no difference to the state, so

5.3

Thus, by direct application of 4.2 and 5.1

5.4

Comparison of 5.2 with 5.4 gives

and 5.5

N 0∈

* N

* ∞

* N * n

n 0=

N

⊕=

*0

x| ∀ *0∈ x| :* 1 * 2→ f | ∀ * 1∈

x| : f | x| f | → x y;| 1

2------- x| f | ,( ) κ f | x| ,( )+[ ]= =

κ % ∈

x y;| x y; |

x y, Ν∈∀

x y; x y; | 12--- x x | y y | κ 2 x x | y y | 2κ x y | y x | + +[ ]=

12--- 1 κ 2+( ) 2κδ x y

2+[ ]=

λ∃ % such that∈ x y;| λy x;| =

x y; x y; | λx y; y x; | 12---λ κ x x | y y | κ x x | y y | 1 κ 2+( ) x y | y x | + +[ ]= =

12---λ 2κ 1 κ + 2( )δ xy

2+[ ]=

1 κ 2+ 2λκ = λ 1 κ 2+( ) 2κ =




Hence . . Substituting into 5.5 ;

if , then so ;

if , then so

Definition: Bosons are particles for which , so that the creation operators obey

5.6

Definition: Fermions are particles for which , so that the creation operators obey

5.7

The use of the ket notation for creation operators is justified by the homomorphism defined by

5.8

It is straightforward to check that this is a homomorphism with the scalar product defined by 4.2 . In general

the creation operator is defined by linearity

5.9

It follows immediately that

5.10

Using 5.6 and 5.7 gives

5.11

5.12

Theorem: The Pauli exclusion principle holds for fermions.

The definition of the creation operator extends to by requiring that its action on each par-

ticle of an n particle state is identical, and that it reduces to 5.1 in the restriction of to the space of the

ith particle. Thus

5.13

where for bosons, for fermions, and appears in the i+1th position in the ith term of the

sum. Normalisation is determined from 4.2 by observing that when all are distinct, the right hand sideis the sum of n+ 1 orthonormal vectors . 5.13 is extended and by linearity.

λ2 1= λ 1±=

λ 1–= 1 κ 2+ 2– κ = κ 1–=

λ 1= 1 κ 2+ 2κ = κ 1=

κ 1= x| ∀ *0∈ x|

y∀ Ν∈ x y;| 1

2------- x| y| ,( ) y| x| ,( )+[ ] y x;| = =

κ 1–= x| ∀ *0∈

y∀ Ν∈ x y;| 1

2------- x| y| ,( ) y| x| ,( )–[ ] y x;| –= =

x| | 1

2------- x| | ,( ) κ | x| ,( )+[ ]=

∀ f | *∈ f | :* 1 * 2 ,→ f | x f | x|

x Ν∈∑=

f | g| *∈,∀

f g;| f | g| =

x f | x| y g | y| y Ν∈∑

x Ν∈∑=

x f | y g | x y;| x y, Ν∈∑=

Boson f | g| * 1∈,∀ f g;| g f ;| =

Fermion f | g| * 1∈,∀ f g;| g f ;| –=

x| :* n * n 1+→* n

x y i,∀ Ν i,∈ 1 … n, ,=

x| : y 1 | …y n| , ,( ) 1n 1+

---------------- x| y 1 | …y n| , , ,( )

κ y i| y 1 | …x| … y n| , , , , ,( )i 1=

n

∑+

→

κ 1= κ 1–= x| x y i,

∀ f | *∈ ∀ g| *∈n




Definition: The space of physically realisable states is the subspace generated from

by the action of creation operators (more strictly physical states are generated by interaction

operators which will composed of creation and annihilation operators).Definition: Notation for the elements of ( is defined inductively.

5.14

Corollary: is identified with the creation operator given by

Definition: The bra corresponding to is .

Theorem:

5.15

where the sum runs over permutations π of (1,2,...,n), ε(π)is the sign of π for fermions and ε(π)=1 for bosons .

Proof: By induction, 5.15 holds for n = 2, by 5.6 and 5.7 . Now suppose that 5.15 holds , Then,from definition 5.14

by the inductive hypothesis. 5.15 follows from application of 5.13 .Corollary: the creation operators obey the (anti)commutation relations

5.16

where, for fermions and bosons respectivelyand

Proof: By definition 5.14 , ,

by 5.15

But by definition the kets span ( . So by linearity

5.16 follows from 5.10 .Theorem:

5.17

( ( Ν( ) * Ν( )⊂=

* 0 | { }=

∀ g| * 1∈ ,∀ f | *∈ n ) ∩ g f ;| g| f | * n 1+ ) ∩∈=

g f ;| * * → g f ;| g| f | =

g f ;| * n 1+∈ g f ; |

x i| ∀ *01

i,∈ 1 … n, ,=

x 1 x 2 … x n;;;| 1

n!--------- ε π( )

π∑ x π 1( )| …x π n( )| , ,( )=

n m 0∈<∀

x 1 x 2 … x m;;;| x 1| x 2 … x m;;| =

1

m 1–( )!------------------------ ε π( ) x 1| x π 2( )| …x π m( )| , ,( )

π∑=

∀ g| , f | *∈

g| f | ,[ ] ± 0=

x y,[ ]+ x y,{ } xy yx+= = x y,[ ] – x y,[ ] xy yx–= =

x i∀ Ν i,∈ 1 … n, ,= x| y| ,∀ *0∈

x| y| , x 1 x 2 … x n;;;| x y; x; 1 x 2 … x n;;;| =

κ y x; x; 1 x 2 … x n;;;| =

κ x| y| , x 1 x 2 … x n;;;| =

x 1 x 2 … x n;;;| x| y| ,[ ] ± 0=

x i| y i| ,∀ *0 i,∈ 1 … n, ,=

y 1 … y n;; x 1 … x n;; | ε π( )π∑ y i x π i( ) |

i 1=

n

∏=




Proof: By 5.15 and 4.2

where we observe that . 5.17 follows since thesum over π' contains n! terms which are identical up to the ordering of the factors in the product.

Corollary:

5.18

Proof: By linearity, 5.9 , and definition 5.14Theorem: is an isomorphic embedding under the

mapping given by

5.19

Proof: By 5.17 and 4.1

since and using 5.17 again.

6 Annihilation OperatorsIn an interaction particles may be created, as described by creation operators, and particles may change

state or be destroyed. The destruction of a particle in interaction is described by the action of an annihilation

operator. A change of state of a particle can be described as the annihilation of one state and the creation of

another, so a complete description of any process in interaction can be achieved through combinations of

creation and annihilation operators. Annihilation operators incorporate the idea that it is impossible to tell

which particle of given type has been destroyed in the interaction. They are defined by their action on a basis

of * , and their relationship to creation operators will be determined. The use of bras to denote annihilationoperators is justified by the obvious homomorphism defined below in 6.2 with . Certain proofs of a

routine nature have been omitted.

Definition: the annihilation operator is given

by

6.1

6.2

y 1 … y n;; x 1 … x n;; | 1n!----- ε π'( )

π'∑ ε π''( )

π''∑ y π' i( ) x π'' i( ) |

i 1=

n

∏=

1n!----- ε π'( )π'∑ ε ππ'( )ππ'∑ y

π' i( ) x

ππ' i( ) | i 1=

n

∏=

permutations π'' π' a permutation π such that π''∃, ,∀ ππ'=

∀ g i| , f j| * i j, ,∈ 1 … n, ,=

g 1 … g n;; f 1 … f n;; | ε π( )π∑ g i f π i( ) |

i 1=

n

∏=

n∀ 0 such that 0 n ( * n∩( ) ( * n 1+∩( )⊂,<,∈

* n * n 1+→ x i∀ Ν i,∈ 1 … n, ,= x 1 … x n;;| | x 1 … x n;;| → x; 1 … x n;;| =

y 1; … y n;; x 1; … x n;; | ε π( )π 1≠∑ | y i x π i( ) |

i 2=

n 1+

∏=

y 1 … y n;; x 1 … x n;; | =

| 1=

n 1=

x| ∀ *0∈ x |:* n * n 1–→ x |: f | x f | * n 1–

∈→

x i∀ Ν i,∈ 1 … n, ,=

x | | x | =

x | x 1| …x n| , ,( ) 1

n------- κ i x x i |

i 1=

n

∑ x 1| … x i 1–| x i 1+| …x n| , , , , ,( )=




The normalisation in 6.2 is determined by observing that when all are distinct, the right hand side is

the sum of n orthonormal vectors. for bosons and for fermions, and is determined by con-

sidering the result of the annihilation operator on a state of one particle in , which is identical

for all values of n under the isomorphic embedding of 5.19 . The annihilation operator for any ket is defined

by linearity

is given by 6.3

Lemma:

6.4

Proof: This is 6.2 withTheorem:

6.5

Proof: By 5.15

by 6.2 , since for each value of there are n permutations π which are identical apart from the

position of i. 6.5 follows by applying 5.15 again.

Theorem:6.6

Proof: By induction, left to the reader.Corollary:

Proof: Immediate from 6.6, by linearity. Hence, it is consistent to define:

Definition: the annihilation operator is given by

6.7

Definition: On a complex vector space, 8 , with a hermitian form, the hermitian conjugate of

the linear operator is defined such that . .

Definition: the creation operator is the hermitian conju-

gate of the annihilation operator, .

6.8

x x i,κ 1= κ 1–=

* 1 *⊂n

) ∩

∀ f | *∈ f |: ) ) → f | f x | x | x Ν∈∑=

x| x 1| x 2| , ,∀ *01

∈

x | x 1| x 2| ,( ) 1

2------- x x 1 | x 2| κ x x 2 | x 1| +=

n 2=

y| x i| ,∀ *0 i,∈ 1 … n, ,=

y x 1 … x n;; | κ i y x i |

i 1=

n

∑x 1 …; x i 1–; x i 1+; … x n;;| =

y |x 1 … x n;;| y |1n!

--------- ε π( )π∑ x π 1( )| …x π n( )| , ,( )=

1

n------- 1

n!--------- κ i y x i |

i 1=

n

∑ n ε π( )π i≠∑ x π 1( )| … x π n( )| , ,( )=

i 1 … n, ,{ }∈

xi

| yi

| ,∀ * 0 i,∈ 1 … n, ,= y n |…y 1 |x 1 … x n;;| y 1 … y n;; x 1 … x n;; | =

x i| ∀ *0 i,∈ 1 … n, ,= f | ∀ * n ( ∩∈

x 1 … x n;; f | x n |…x 1 | f | =

x i| ∀ *0 i,∈ 1 … n, ,= x 1 … x n;; |:* * →

x 1 … x n;; | x n |…x 1 |=

φ†:8 8 →

φ:8 8 → f g 8 ∈,∀ φ† f g,( ) f φg,( )=

x i| ∀ *0 i,∈ 1 … n, ,= x 1 … x n;;| :( ( →

x 1 … x n;; |:( ( →

x 1 … x n;; |† x 1 … x n;;| =




Proof: From the definition, ,

by applying 6.7 three times. Thus is the map

which demonstrates 6.8 .Corollary: the annihilation operators obey the (anti)commutation relations.

6.9

Proof: Straightforward from, 5.16 , the (anti)commutation relations for creation operators.Theorem: the creation operators and annihilation operators obey the (anti)commutation

relations

6.10Proof: From 6.5 and using linearity, left to the reader

7 InteractionsIn this treatment (ý is simply a labelling system and its construction has required no physics beyond the

knowledge that we can measure the position of individual particles, and that we can measure the relative

frequency of each result of a repeated measurement. The description of physical processes in terms of thislabelling system requires a law describing the time evolution of states. Let be a finite discrete time

interval such that any particle under study certainly remains in N for . Without loss of generality let. An interaction at time t is described by an operator, I (t ):( → ( . For definiteness we may take

, 7.1

since otherwise there would be a component of I corresponding to the absence of interaction. At each time

t, either no interaction takes place and the state is unchanged, or an interaction, I , takes place. By

the identification of the operations of vector space with weighted OR between uncertain possibilities, the

possibility of an interaction at time t is described by the map

where µ is a scalar chosen to preserve the norm, as required by the probability interpretation. Thus the law

of evolution of the ket from time t to time is7.2

It is straightforward to see that 7.2 is in some sense approximated by the time evolution equation

found in, for example, lattice field theory [13] . Lattice field theory does not describe

the model of simple particle interactions considered here, so there is motivation for a somewhat modified

treatment. Observe that Hilbert space is information space at some time t , , so the interaction is

x i y j,∀ Ν i,∈ 1 … n j, , , 1 … m, ,= = f | ) ∈∀

y 1 … y n;; |x 1 … x n;; |† f | y 1 … y n;; |x 1 … x n;; | f | =

y n |…y 1 |x n |…x 1 | f | =

x 1 … x n;; y; 1 … y n;; | f | =

x 1 … x n;; |† x 1 … x n;; |† : y 1 … y n;;| x 1 … x n;; y; 1 … y n;;| →

∀ g| , f | *∈

g | f |,[ ] ± 0=

∀ g| , f | *∈

g | f | ,[ ] ± g f | =

Τ 0⊂

x0 Τ∈Τ 0 T ),[=

x i∀ Ν n 0∈∀,∈ x 1 … x n;; | I x 1 … x n;;| 0=

f | ) ∈

) → f | µ1 iI t ( )–( ) f | →

t 1+ f | t 1+ µ 1 iI t ( )–( ) f | t =

f | t 1+ µe iI t ( )– f | t =

( ( t ( )=




a map

Since is a map from one Hilbert space to another we cannot talk of it being self adjoint, or of the spec-trum of . Although we can identify the Hilbert spaces at different times using the natural

homomorphism defined by the basis of position kets, when we do so we cannot use a linear operator

on the resultant space to physically represent an interaction. Linearity is normally imposed since whenacts on a state prior history should not be relevant. But a more careful analysis suggests that is not

strictly linear, because linearity would dictate that an the action of on a particle created by at time

t should be the same as its action on a particle previously created and evolving to the same ket, so that a

particle can physically interact twice in the same instant. This appears physically meaningless and we reg-

ularise by imposing the condition that a particle cannot be annihilated at the instant of its creation. In

other words cannot act on the results of itself. This regularisation only makes sense if the unit, χ,

describing one instant of time is actually a fundamental physical parameter of the universe equal to latticespacing. Thus we have

7.3

In other respects is linear. The removal of products describing the annihilation of a particle at the instantof its creation, as in 7.5 is most naturally done by normal ordering. In practice this is largely academic, since

when we find the perturbation expansion as an iterative solution of 7.2 terms containing are already

excluded. We will see that the exclusion of these terms leads directly to a finite “renormalised” perturbation

expansion. It might be thought that 7.3 would prevent an interaction taking place in each instant, and so

would prevent interaction altogether but this is only the case if there is also an observation in each instant.

We may regard this as a limiting instance of a quantum Zeno effect, which is known in quantum mechanics

to stop interaction under conditions of continuous observation [10]

By 7.2 preservation of the norm implies that

7.4

7.5

Normal ordering implies that So

7.6

7.6 has a straightforward solution with . and . Although strictly non-linearity implies that I

is not hermitian, does not appear in the physical model, as discussed above, and we may treat I as her-

mitian and will refer to it as such. 7.2 can be interpreted literally as meaning that in each instant particleeither interacts or does not interact. In the latter case the state remains the same and is multiplied by a phase,

, so that 7.2 reduces to

7.7

7.7 is a geometric progression with solution

7.8

I t ( );( t ( ) ( → t 1+( ) I t ( )

1 iI t ( )– I t ( ) I t ( )

I t ( ) I t ( ) I t ( )

I t ( ) I t ( )

I 2 t ( ) 0=

I t ( )

I 2 t ( )

f | ( ∈∀

f f | f |1 iI †+( )µµ1 i– I ( ) f | =

µ 2= f f | f | I † I f | i f | I † I – f | + +( ) f | I † I ( ) f | 0=

f | I † I – f | f f | ----------------------------- i 1 µ 2–

µ 2-----------------=

µ 2 1= I I †=

I 2 t ( )

µ e i– E = E 4 ∈

f | t 1+ e i– E f | t =

f | t e i– Et f | 0=




8 CovarianceClearly the use of a discrete lattice requires that we redefine Lorentz transformation. The lattice is invar-

iant because transformation introduces a new lattice, aligned with the new coordinate axes and retaining theparameters χ and ν. ν is an arbitrary observer choice. If lattice size, ν, is changed then momentum space is

redefined; this must be done in co-ordinate space. The same is true of any change of co-ordinate system,

such as to curvilinear co-ordinates. As seen in section 7, for a useful time evolution equation χ is a funda-mental physical constant, and is the same in all reference frames. We will continue to use . We onlyrequire local invariance as in general relativity, and we seek to carry out transformations only when all the

matter under study is contained in the reference frames both before and after transformation. We require a

proscription relating predictions made in one lattice with predictions made in another. From 3.6 the generalsolution of 7.7 at time t is

8.1

Phase, , is arbitrary, but we seek a covariant solution and we fix µby defining:

Definition: is the time like component of a vector . E is called energy.

Only the integrand in 8.1 is covariant. The integral operator is defined in an identical manner in all ref-

erence frames, and is invariant. It will be found that E is conserved in measured states and can be identified

with classical energy. By definition we have a mass shell condition where m is a constant,

known as bare mass. It follows immediately that elementary particles obey the Klein-Gordon equation. In

this treatment the Klein-Gordon equation is an identity based on the definition of energy as the time com-

ponent of an equation of motion, and will not be treated as an equation of motion.Although is discrete in x and t , on a macroscopic scale it appears continuous. 8.1 can be

embedded into a continuous function , called the wave function defined by

8.2

So we have

8.3

8.2 is not manifestly covariant, but under reasonable conditions it is invariant for physically realisable states

and transformations. To see this we observe that since energy is conserved in physical measurement momen-

tum is always bounded by the total energy of a system (it may not be strictly possible to bound the support

in both momentum space and coordinate space, but it is possible to do so to the accuracy of experiment).

The probability of finding a momentum above this bound is zero, and we assume that for physically realis-able states the discrete representation of has support which is bounded below in each

component of momentum. The bound depends on the system under consideration, but without wishing tospecify it, we observe that, provided that the discrete unit of time is sufficiently small, it is always much less

than . Physically meaningful Lorentz transformation cannot boost it beyond this value because physi-cal reference frames are defined with respect to macroscopic matter. A realistic Lorentz transformation

means that macroscopic matter has been physically boosted by the amount of the transformation. If lattice

spacing is a boost in the order of π would require an energy of solar masses per kilogram

χ 1=

x t , f | 12π------ 3

2---

d 3 p p f | e i– Et x p⋅–( )

Μ∫ =

µ e iE =

E p 0= p E p,( )=

m2 E 2 p 2–=

x t , f | f :4 4 % →

x 4 4∈∀ f x( ) 12π------ 3

2---

d 3 p p f | e ix p⋅–

Μ∫ =

x Ν t ∀ Τ x f | ∈,∈∀ t x, f | f t x,( ) f x( )= = =

p f | π χ ⁄

π χ ⁄

Gm e c 3 ⁄ 9 1014×




of matter to be boosted, which may be thought unrealistic. Thus we can ensure covariance by imposing the

condition that all physical momentum space wave functions have a representation in a subset of Μbounded

by some realistic energy level (this bound will not affect the calculation of loops in Feynman diagrams, since

these are not observable states). Then we remove the non-physical periodic property of by replacing

where if and otherwise. 8.2 is then replaced with

8.4

For any ket, there is a unique momentum space function defined by 3.4 , and a unique wave func-

tion defined by 8.4 . So in each coordinate system there is a homomorphism between * and thevector space of wave functions with the hermitian product defined by 2.5

8.5

Clearly Lorentz transformation cannot be applied directly to a discrete co-ordinate system, but it can be

applied to the wave function, 8.4 . Then 8.3 defines the position function, and hence a ket, by the restrictionof the wave function to the transformed co-ordinate system, , at integer time. Let be a state of

a particle at definite position x in the lattice at some time . Then, from 3.11 , the Lorentz transformation is

So we have for Λ

8.6

We impose a new co-ordinate system at time t' after transformation by restricting the wave function to points x' in a new cubic lattice Ν'. Then the transformed state is the restriction to the new lattice, i.e

8.6 gives

is not an eigenstate of position in Ν'; if a measurement of position were done in Ν' and we were then

to transform back to Νthe state would no longer be . Thus the operators for position in different frames

Νand Ν' do not commute. But if no measurement is done, we can transform straight back and recover ,

showing that there is no problem with lack of unitarity under Lorentz transformation.Proof:

p f | ΘΜ p( ) p f | p f | →

ΘΜ p( ) 1= Μ∈ ΘΜ p( ) 0=

f x( ) 12π------ 3

2---

d 3 p p f | e ix p⋅–

4 3∫ =

p f | Ν 0 ∈

g f | g x( ) f x( ) x Ν∈∑=

Ν′ 0 ∈ x| x0

Λ x| d 3 p

4 3∫ p| p Λ x | d 3 p

4 3∫ p| Λ′p x | = =

d 3 p

4 3∫ Λ p| p | Λ d 3 p

4 3∫ p| Λ′p |= =

Λ x| x ′| x′ Λ x | Ν′∑=

Λ x| d 3 p x ′| x′ p | Ν′∑

4 3∫ Λ′ p x | =

Λ x| x|

x|

Λ′ Λ y| d 3 p Λ′4 3∫ x ′| x′ p | Λ′p y |

x Ν′∈∑=




Restrict to the original coordinates

9 Wave MechanicsWave functions are not restricted to ý.ý 2, but, in any reasonable definition of an integral, 8.5 is approxi-

mated by the hermitian product in .ý 2 whenever f and g are in .ý 2 and the spacing of the lattice is small.

The law for the time evolution of the wave function is given by differentiating 8.2

9.1

7.7 is obtained by integrating 9.1 over one time interval. Thus, in the restriction to integer values, 9.1 is iden-

tical to 7.7 , the difference equation for a non-interacting particle. It is therefore an expression of the samerelationship or law. As an equation of the wave function, the right hand side of 9.1 is a scalar, whereas the

left hand side is the time component of a vector whose space component is zero. So 9.1 is not manifestly

covariant. A covariant equation which reduces to 9.1 requires the scalar product between the vector deriv-ative, and the wave function and has the form, for some vector operator Γ

9.2

Then the time evolution of the position function in any reference frame Ν is the restriction of the solution

of 9.2 to Νat time . As discovered by Dirac [4] , there is no invariant equation in the form of 9.2 for

scalar f and the theory breaks down. To rectify the problem a spin index is added to Ν

where S is a finite set of indices. When there is no ambiguity we write , and the constructions of

the vector spaces, * , * and ( , go through as before.

When we wish to make the spin index explicit we write normalised by 2.1

9.3

The wave function acquires a spin index

9.4

and the braket becomes

9.5

Λ′ Λ y| d 3 p

4 3∫ x| x Λ′ x′ | x ′ p | Λ′p y |

x ′ Ν′∈∑

x Ν∈∑=

d 3 p

4 3∫ x| Λ x x′ | x ′ Λ p | p y |

x ′ Ν′∈

∑ x Ν∈∑=

d 3 p4 3∫ x| Λ x Λ p | p y |

x Ν∈∑=

d 3 p4 3∫ x| x p | p y |

x Ν∈∑=

y| =

i f 0∂ E f =

,∂i Γ ⋅ f ∂ m f =

t Τ∈

ΝS Ν S⊗= for v 0∈

Ν ΝS=

x| x α,| x| α= =

x α,( ) y β,( ) ΝS∈,∀ x α, y β, | x y | αβ δ xy δαβ= =

f x( ) f α x( ) x f | α= =

g f | g x | x f | x ΝS∈∑ gα x( )

x α,( ) ΝS∈∑ f α x( )= =




It is now possible to find a covariant equation which reduces to 9.1 in the particle’s reference frame,

namely the Dirac equation,

9.6

Another possibility is that f is a vector and that 9.1 is a representation of a vector equation (with )

9.79.7 is understood as the equation of motion of a vector particle which is only ever created or destroyed in

interaction, and for which there is no interval of proper time between creation and annihilation (zero proper

time for the emission and absorption of a light beam is a familiar result from special relativity). The norm

is intended to generate physically realisable predictions of probability, and must be both invariant and pos-itive definite. It is given by

9.8

If f transforms as a space-time vector, 9.8 is only invariant if 9.3 is replaced by the definition

9.9

where η is given by

η(0) = -1 and η(1) = η(2) = η(3) = 1.

We will use the summation convention for repeated spin indices, but not the convention of raising and low-

ering indices. The factor -1 is implicit in summing the zeroeth index for vectors, so 9.5 and 9.8 are retained.

9.9 is invariant, but not positive definite, as required by a norm. The definition of the braket in terms of prob-

ability implies that any vector particles have a positive definite norm for physical states, so only space-like

polarisation of massless vector particles can be observed physically. Other states are permitted, and are

required if interactions are to give correct physical predictions, but we can only discuss the probability of

observing them if there is positive definite norm.

10 Dirac ParticlesThe solution to 9.6 is

10.1

where p satisfies the mass shell condition and u is a Dirac spinor, having the form

for r = 1,2 10.2

where ζ is a two-spinor normalised so that

10.3

and σσσσ = (σ1,σ2,σ3) are the Pauli spin matrices.

i ∂ γ f x( )⋅ mf x( )=

Γ 1=

i ∂ f x( )⋅ 0=

f f | f α x( )x α,( ) ΝS∈∑ f α x( )=

x α,( ) y β,( ) Ν∈,∀ x α, y β, | x y | αβ η α( )δ xy δαβ= =

f α x( ) 12π------ 3

2---

d 3 p F p r ,( )uα p r ,( ) e ix p⋅–

4 3∫

r 1=

2

∑=

u p r ,( ) p 0 m+2 p 0

----------------ζ r ( )

σσσσ p⋅

p 0 m+---------------- ζ r ( )

=

ζα r ( )ζα s( ) δrs=




It is routine to show the spinor normalisation

F ( p ,r ) is the momentum space wave function given by inverting 10.1 at

10.4

Definition: p0 is the energy of a state with momentum p . p = (p 0 , p ) is called energy-momentum; p 0 will later

be identified with classical energy.

Definition: With the Dirac γ -matrices as defined in the literature the Dirac adjoint is

Lemma: The γ -matrices obey the relations

and 10.5

Proof: These are familiar matrix equations and the proof is left to the reader

Lemma: In this normalisation Dirac spinors obey the following relations

10.6

10.7

Proof: These are familiar spinor relations renormalised and the proof is left to the reader.This normalisation is consistent the definition of ket space in the reference frame of an individual

observer and leads to some simplification of the formulae. Wave functions are non-physical and it is notnecessary to use the invariant integral.

The most fundamental representation of the discrete equation of motion, 7.2 , is most readily understood

as describing the interactions of a particle in the proper time of that particle. This being so there is no way

to say that a particles proper time cannot become reversed with respect to macroscopic matter. The treatment

of the antiparticle modifies the Stückelberg-Feynman [19] ,[6] interpretation by considering the mass shell

condition. A sign is lost in the mass shell condition, due to the squared terms, but a time-like vector with a

negative time-like component has a natural definition of m < 0. So permissible solutions of the Dirac equa-tion, 9.6 , have positive energy when m is positive and negative energy when m is negative.

Complex conjugation reverses time while maintaining the probability relationship, 2.2 , and restores positiveenergy, and we also change the sign of mass, . Thus, given that no interaction takes place, the ket

for a Dirac particle in its own reference frame evolves according to 7.8 , for both and . But the

negative energy solution is transformed and satisfies

10.8

where γ is the complex conjugate, . Although this is a slightly different from the positron wave

function cited in e.g. [1] the treatments will be reconciled in the definition of the field operators. The solution

u a p r ,( )uα p s,( ) δrs=

x0 0=

F p r ,( ) χ2π------ 3

2---

f α

0 x,( )uα

p r ,( )e i x p⋅

x α,( ) Ν∈∑=

u uγ 0=

γ 0γ 0 1= γ 0γ α†γ 0 γ α=

p γ ⋅ m–( )u p r ,( ) 0 u p r ,( ) p γ ⋅ m–( )= =

uα p r ,( )uα p s,( ) δrs m p 0-----=

uα p r ,( )uβ p r ,( )r 1=

2

∑ p γ ⋅ m+2 p 0

--------------------

αβ=

E p 0 0>=

m m–→m 0> m 0<

i ∂ γ f x( )⋅ m– f x( )=

γ αβ j γ αβ

j=




to 10.8 is the wave function for the antiparticle

10.9

where p satisfies the mass shell condition, and is the complex conjugate of the Dirac spinor.

for r = 1,2

10.9 is the complex conjugate of the negative energy solution of the Dirac equation. The spinor has thenormalisation

F ( p ,r ) is the momentum space wave function given by

10.10

Lemma: In this normalisation the Dirac spinors obey the following relations

10.11

10.12

Proof: These are familiar spinor relations renormalised and the proof is left to the reader.

11 The PhotonWe require a solution to the equation of motion, 9.7 . By definition of energy ( section 8 ) every particle

obeys a Klein-Gordon equation. Positive definite norm implies that the only possibility has zero mass (mas-

sive vector bosons are allowed but no probability amplitude exists for them and only their decay products

may be directly observed). The wave function for the photon is

11.1

where

i. p2 = 0 (from the Klein-Gordon equation with zero mass)ii. w are orthonormal vectors given by

a) time-like component:b) space-like components: for r = 1,2,3 are such that is lon-

gitudinal and so w ( p ,1) and w ( p ,2) are transverse

f x( ) 12π------ 3

2---

d 3 p F p r ,( )v p r ,( ) e ix p⋅–

4 3∫

r 1=

2

∑=

v

v p r ,( ) p 0 m+

2 p 0-----------------

σσσσ p⋅

p 0 m+---------------- ζ r ( )

ζ r ( )=

vα p r ,( )vα p s,( ) δrs=

F p r ,( )1

2π------ 3

2---

f α 0 x,( )vα p r ,( )ei x p⋅

x α,( ) Ν∈∑=

p γ ⋅ m+( )v p r ,( ) 0 v p r ,( ) p γ ⋅ m+( )= =

vα p r ,( )vα p s,( ) δrs m p 0-----=

vα p r ,( )vβ p r ,( )r 1=

2

∑ p γ m–⋅2 p 0

--------------------

αβ=

f α x( ) χ2π------ 3

2---

d 3 p F p r ,( )wα p r ,( ) e ix p⋅–

4 44 4 3

∫ =

w p r ,( ) 1 0,( )=

w p r ,( ) 0 w p r ,( ),( )= w p 3,( ) p p 0 ⁄ =

w p r ,( ) w p s,( )⋅ δrs=




iii. F is such that the photon cannot be polarised in the longitudinal and time-like spin states, i.e.

F ( p ,0) = F ( p ,3) 11.2

Proof: With the above definitions

p.w ( p ,3) = p0 = - p.w ( p ,0) and p.w( p ,1) = p.w ( p ,2) = 0

So that differentiating 11.1

= 0

by 11.2 . This establishes that 11.1 is the solution to 9.7F ( p ,r ) is the momentum space wave function given by inverting 11.1 at

11.3

12 Plane Wave StatesDefinition: plane wave states are defined by the wave functions

for the Dirac particle 12.1

for the antiparticle, and 12.2

for the photon. 12.3

Theorem: (Newton’s first law) In an inertial reference frame, an elementary particle in isolation has a con-stant momentum space wave function.

12.4

Proof: Clearly plane waves are solutions of 9.6 , 10.8 and 9.7 so they describe the evolution of states in iso-

lation. For each of the Dirac particle, antiparticle and photon, by 2.6 , ,

12.5

Substituting 12.1 , 12.2 and 12.3 in 12.5, with x0 = 0, and examining 10.4 , 10.9 and 11.3 reveals

for the Dirac particles, and 12.6

for the photon. 12.7

i f ⋅∂ x( ) 12π------ 3

2---

d 3 p F p r ,( ) p w α p r ,( ) e ix p⋅–⋅

4 3∫

r 0=

3

∑=

12π------ 3

2---

= d 3 p p 0 F p 3,( ) F – p 0,( )( )e ix p⋅–

4 3∫

r 0=

3

∑

x0 0=

F p r ,( ) χ2π------ 3

2---η r ( ) f α 0 x,( )wα p r ,( )e i x p⋅

x α,( ) Ν∈∑=

x0 Τ∈∀ p r ,| *=

x p r , | 12π------

32---u p r ,( )e ix p⋅–=

x p r , | 12π------

32---v p r ,( )e ix p⋅–=

x p r , | 12π------

32---w p r ,( )e ix p⋅–=

f | ∀ *∈

f | ηr ( ) d 3 p

4 3∫

r ∑ p r ,| p r , f | =

f | ∀ *∈ x0 Τ∈∀

p r , f | p r , x | ' x f | x Ν∈∑=

p r , f | F p r ,( )=

p r , f | ηr ( )F p r ,( )=




Corollary: The time evolution of the position function of a particle in isolation is,

12.8

where r = 0-3 for photons, and r = 1-2 for Dirac particles ( η is redundant for a Dirac particle).

Proof: Substituting 12.6 and 12.1 into 10.1, 12.6 and 12.2 into 10.9 , and 12.7 and 12.3 into 11.1 gives, ineach case, 12.8 .Corollary: The resolution of unity

12.9

Proof: 12.4 is true for all .Corollary: The braket has the time invariant form

12.10

Proof: Immediate from 12.9Theorem: is a delta function on the test space of momentum space wave functions

12.11

Proof: From 12.10 , for plane wave

Corollary: The braket for the photon is positive definite, as required by the probability interpretation.

Proof: By 11.2 and 12.7 the time-like ( r = 0) and longitudinal ( r = 3) states cancel out in 12.10 and for pho-

tons as well as Dirac particles 12.10 reduces to

12.12

Theorem: (Gauge invariance). Let g be an arbitrary solution of . Then observable results areinvariant under gauge transformation of the photon wave function given by

12.13

Proof: It follows from 12.12 that the braket is invariant under the addition of a (non-physical) light-likepolarisation state, known as a gauge term. Let be an arbitrary function of momentum. The general

solution for g is

where . Then

is equivalent to a light like polarisation states, and has no effect on the braket. is known as a gauge term,

f | ∀ *∈

x f | ηr ( ) d 3 p

4 3∫

r ∑ x p r , | p r , f | =

η r ( ) d 3 p

4 3∫

r ∑ p r ,| p r , | 1=

f | *∈

g f | ηr ( ) d 3 p

4 3∫

r ∑ g p r , | p r , f | =

q s, p r , | q s, p r , | ηr ( )δrs δ p q–( )=

q s,|

q s, f | ηr ( ) d 3 p

4 3∫

r ∑ q s, p r , | p r , f | =

g f | d 3 p

4 3∫

r 1=

2

∑ g p r , | p r , f | =

g2∂ 0=

f α x( ) f α x( ) gα x( )∂+→

G p( )

g d 3 p4 3∫ e i– p x⋅ G p( )=

p 2 0=

gα∂ d 3 p

4 3∫ p 0 w p 0,( ) w p 3,( )+( )e i– p x⋅ G p( )=

gα∂




and has no physical meaning. It follows from 12.12 that light-like polarisation cannot be determined from

experimental results. Although their value is hidden by the gauge term, the time-like and longitudinal polar-

isation states cannot be excluded, and we will see that they contribute to the electromagnetic force.Theorem: Space-time translation by displacement, z, of the co-ordinate system such that the particleremains in Ν,is equivalent to multiplication of the momentum space wave function by .

Proof: Using 12.6 and/or 12.7 in 12.4 .

12.14

Under a space-time translation, z, by 12.1 , 12.2 and 12.3 we have,

12.15

as required.

13 Field Operators

Definition: A partial field is a family of mappings , where S is the set

of spin indices introduced in section 8 , the elements of ( (Ν) are regarded as operators.

Definition: The partial field of creation operators for a particle in interaction is

. 13.1

We will find that photons are not created in eigenstates of position so we do not in general have

. will be found for each particle.

Definition: Let be the operator for the creation a particle at the origin.Definition: By 6.8, the annihilation operator is the hermitian conjugate.

Theorem: The creation operator for a particle at is given by

13.2

Proof: By the resolution of unity, 12.9 , is given by

13.3

The momentum space bound has to be restored because we do not have a momentum space wave functionwith bounded support, but an operator on Hilbert space. By the principle of homogeneity space-time trans-

lation maps the creation operators appearing in interactions into each other. Then, by 12.15 ,

13.4

13.2 follows by substituting 13.4 into 13.3.

e ip z⋅

x f | d 3 p

4 3∫

r ∑ F p r ,( ) x p r , | =

x z– f | d 3 p

4 3∫

r ∑ F p r ,( )e ip z⋅ x p r , | =

Ψ Ψ Ν( )= Ψ Ν( ):4 4 S⊗ ( Ν( )→

x α,|

x α,( )∀ x0 x α, ,( ) 4 4 S⊗∈= x α,| :( ( →

x Ν x α,| ,∈∀ x α,| = x α,|

α| 0 α,| = x α, |:( ( →

x α,| :( ( → x α,( ) 4 4 S⊗=

x α,| ηr ( ) d 3 p

Μ∫

r ∑ p r , α | e ip x⋅ p r ,| =

x α,| :( ( →

x α,| ηr ( ) d 3 p

Μ∫

r ∑ p r , x α, | p r ,| =

p r , x α, | p r , α | e ip x⋅=




Definition: The derivative of the creation and annihilation operators is defined by differentiating 13.2 .

13.5

There may be a number of different types of interaction, described by I j: ( → ( , where j runs over an

index set. Let be the coupling constant for the interaction I j. Only one type of interaction takes placeat a time, but there is uncertainty about which. Under the identification of addition with quantum logical OR,

the interaction operator I ( x0): ( → ( , introduced in section section 7 , is

I is hermitian, and each I j is independent by definition, so each I j is hermitian 9 (up to regularisation).Definition: In any finite discrete time interval, Τ , for each type of interaction, an operator,

H ( x): ( →( ,

describes the interaction taking place at , H ( x) is called interaction density.

The principle of homogeneity implies that H ( x) is the same, up to homomorphism, and has equal effecton a matter anywhere in N and for all times in T. I j describes equal certainty that a particle interacts any-where in Ν, so by the identification of addition with quantum logical OR , can be written as a sum

13.6

The sum in 13.6 is over space, but not necessarily over the spin index. Without loss of generality H ( x) ishermitian. By the definition of multiparticle space as a direct product (section 4), H ( x) can be factorised as

a product of Hermitian operators, J γ ( x), where γ runs over the particles in the interaction

13.7

Definition: J is called a current operator (its relationship to the electric current will be shown) .

A number of particles participate in the interaction. As described by operators, the particles prior to inter-action are annihilated and the particles present after interaction are created – a particle which is physically

preserved is described as being annihilated and re-created. H ( x) can be represented as a Feynman node. Each

line at the node corresponds to one particle in the interaction. In a single Feynman node there are no geo-metrical relationships with other matter, so it is not possible to say whether a particle’s proper time is

running forwards or backwards with respect to the reference frame clock. So a line for the annihilation of a

particle, γ , may also represent the creation of the corresponding antiparticle .

Definition: Let be the annihilation operator for a particle at , and let

be the creation operator for the antiparticle. Then the particle field :( →( is

13.8

Then each line at the Feynman node corresponds to a particle field modelling the creation or annihilation of

a particle. Clearly the hermitian conjugate of a particle field is the antiparticle field

13.9

x α,| ∂ x α,∂| ηr ( ) d 3 p

Μ∫

r 0=

3

∑ p r , α | ipe ip x⋅ p r ,| = =

e j 4 ∈

I e j I j j∑=

x x0 x,( ) Τ Ν⊗∈=

I j

I j x0( ) H x 0 x,( ) x Ν∈∑ H x( )

x Ν∈∑= =

H x( ) J γ x( )γ ∏

=

γ

x α, | x α,( ) x0 x α, ,( ) Τ Ν⊗∈=

x α,| φα x( )

φα x( ) x α,| x α, |+=

φ†α x( ) x α,| x α, |+=




In the general case J γ ( x) is hermitian so it combines the particle and antiparticle fields

13.10

Then the general form of the interaction is

13.11

The colons reorder the creation and annihilation operators by placing all creation operators to the left of all

annihilation operators, to ensure that false values are not generated corresponding to the annihilation of par-

ticles in the interaction in which they are created. Particular interactions can be postulated as operators with

the general form of 13.11 , we can examine whether the resulting theoretical properties correspond to theobserved behaviour of matter.

Definition: Let π be the permutation such that Then the time ordered product is

Theorem: (Locality)

such that is space-like 13.12

Proof: Iterate 7.2 from an initial condition at t = 0 given by

Expand after T iterations

13.13

Then 13.13 is

13.14

There may be any number of particles in the initial state so 13.14 can be interpreted directly as

a quantum logical statement meaning that, since an unknown number of interactions take place at unknownpositions and unknown time, the final state is (named as) the weighted sum of the possibilities. Except for

asymptotically free initial and final states, this statement ceases to make sense in the limit , whichforces to ensure that particles remain in Ν. The expansion may reasonably be expected not

to converge under such conditions, but there is no problem for bounded Νand finite values of T (i.e. stable

J γ x( ) J γ φα x( ) φ†α x( ),( )=

I j x0( ) : J γ x α,| x α, | x α,| x α, |+,+( ):γ ∏ x Ν∈∑

=

τπ n( ) …τπ 2( ) τ> π 1( )>

T I τn( )… I τ1( ){ } I τπ n( )( )… I τπ 1( )( )=

x y Τ Ν⊗∈,∀ x y– H y( ) H x( ),[ ] 0=

f | 0∈

f | 1 µ 1 iI 0( )–( ) f | 0=

f | 2 µ2 1 iI 1( )–( ) 1 iI 0( )–( ) f | 0=

f | 3 µ3 1 iI 2( )–( ) 1 iI 1( )–( ) 1 iI 0( )–( ) f | 0=

f | T µT 1 i I τ( )τ1 0=

T 1–

∑ i–( )2 I τ2( ) I τ1( )τ1 0=

T 1–

∑ …+τ2 0=

τ2 τ1>

T 1–

∑+ +

f | 0=

f | T µT 1 i–( )nn!

------------n 1=

T

∑ T I τn( )… I τ1( ){ }τ1…τn 0=

i j τ⇒≠ i τ j≠

T 1–

∑+

f | 0=

f | 0∈

T ∞→ΝS

0→ 3S⊗




fore and after states). By 13.6 , 13.14 is

13.15

Under Lorentz transformation of 13.15 the order of interactions, , can be changed in the time ordered

product whenever is space-like. But this cannot affect the final state for any .Corollary: By 13.7 H factorises and the locality condition applies to the current operators.

such that is space-like 13.16

14 Classical LawTheorem: In an inertial reference frame momentum is conserved.

Proof: Classical momentum is the expectation of the momentum of a large number of particles. In theabsence of interaction the expectation of momentum is constant for each particle by Newton’s first law, 12.4.

So it is sufficient to prove conservation of momentum in each particle interaction. Expand the interactiondensity, 13.11 , as a sum of terms of the form

14.1

where and are creation and annihilation operators for the particles and antiparticles in theinteraction, given by 13.2 . Suppress the spin indices by writing and

. We have from 14.1 , , plane wave

Then, by 5.18

which is a sum of terms of the form

.

Using 13.4 and permuting this reduces to a sum of terms of the form

by 3.12 . Thus momentum is conserved in each particle interaction, and so is conserved universally by New-

ton’s first law 12.4 .

Remark: Conservation of momentum depends solely on the principle of homogeneity as expressed in 13.4 ,

f | T µT 1 i–( )nn!

------------n 1=

T

∑ T H x n( )… H x 1( ){ } x1… xn Τ ΝS⊗∈

i j x0

i x0

j≠⇒≠

∑+

f | 0=

H x i( ) x i x j– f | T T 0∈

x y Τ ΝS⊗∈,∀ x y– J y( ) J x( ),[ ] 0=

i x0( ) h x( ) x Ν∈∑ x α,| 1

x Ν∈∑ … x α,| m x α, |m 1+ … x α, |n= =

x α,| i x α, |i p Μ∈∀ s 1 2 3 4, , ,= p| p s,| =

x| x α,| = n∀ m 0 n m 0>, ,∈, ∀ p 1| …p n| , ,

p 1 … p m;; |i x0( )p m 1+ … p n;;| p 1 …p m;; | x| 1

x Ν∈∑ … x| m x |m 1+ … x |n p m 1+ … p n;;| =

p 1 … p m;; |i x0( )p m 1+ … p n;;| ε π( )π∑ p i x | π i( )

i 1=

m

∏ x Ν∈∑ ε π'( )

π'∑ x p π' j( ) j |

j m 1+=

n

∏=

q i x | π i( )i 1=

m

∏ x Ν∈∑ x p j π' j( ) |

j m 1+=

n

∏ p π' j( ) p j→

q i α | e iq i x⋅

i 1=

m

∏ x Ν∈∑ α p j | e i– p j x⋅

j 1=

n

∏ δ p j

j m 1+=

n

∑ q i

i 1=

m

∑–

q i α | e iq 0i x0– α p j | e p 0

i x0–

j 1=

n

∏i 1=

m

∏=




and the mathematical properties of multiparticle vector space imposed by definition. Energy is not con-

served in an individual interaction.

We are interested in changes in classical observable quantities. That is changes in the expectation,of an observable, . Since measurement is a combination of interactions, all observa-

ble quantities are composed of interaction operators, which, by 13.11 , can be decomposed into fields. Thus

physically observable discrete values are obtained from differentiable functions, and difference equationsin the discrete quantities are obtained by integrating differential equations over one unit of time.Lemma: The equal time commutator between an observable operator O such that and

the interaction density H obeys the commutation relation

14.2

Proof: Immediate from 13.16Theorem: The expectation of an observable operator obeys the differential equations

14.3

Proof: By 7.2

by 7.4, since the state is an eigenstate of O and . Then

Using linearity of kets treated as operators and rearranging14.4

The solution to 14.4 is the restriction to integer values of the solution of

14.5

Using locality, 14.2 , with 14.5 is

14.6

Using locality, 13.12 , 14.6 reduces to the time-component of 14.3 . The proof for is trivial.

Corollary: Particles are point-like.Note: Position is only a numerical value derived from a configuration of matter in measurement, and it is

not obvious that this requires that particles are themselves point-like.Proof: By 14.2 and 14.3 changes in have no dependence on interactions except at the point x.

Corollary: No observable particle effect may propagate faster than the speed of light.

Proof: By 14.3 has no space-like dependence on particle interactions for any space-like slice.

14.2 involves the commutation relation between the interaction density, H , and the observable, O. Since

all physical processes are described by interactions, any observable operator is a combination of interaction

O O O x( ) O t x,( )= =

O x( ) O H x( )( )=

x∀ y H x( ) O y( ),[ ] x0 y0=,≠ 0=

O x( ) O H x( )( )=

O x( ) 0∂ i H x( ) O x( ),[ ] O0∂ x( ) +=

For α 1 2 3, ,= O x( ) α∂ Oα∂ x( ) =

O t 1+( ) f |t 1 iI t 1+( )+( )O t 1+( ) µ2 1 iI t 1+( )–( ) f | t =

f |t = i I t 1+( ) O t 1+( ),[ ] O t 1+( ) f | t +

µ 21=

O t 1+( ) O t ( ) – f |t 1+ O t 1+( ) f | t 1+ f |t O t ( ) f | t –=

f |t = i I t 1+( ) O t 1+( ),[ ] O t 1+( ) f | t f |t O t ( ) f | t –+

O t 1+( ) O t ( ) – i I t 1+( ) O t 1+( ),[ ] O t 1+( ) O t ( )– +=

O x( ) 0∂ i I t ( ) O x( ),[ ] O0∂ x( ) +=

x0 y0=

O x( ) 0∂ i H x 0 y,( ) y Ν∈∑ O x( ), O0∂ x( ) +=

α 1 2 3, ,=

O x( )

O x( )




operators, so observables are a combination of particle fields. Then 14.2 requires the commutators for par-

ticle fields. For fermions the creation operators anticommute, but commutation relations are obtained if the

current, 13.10 , is a composition of an even number of fermion fields.

15 The Photon FieldPhotons are bosons, and having zero mass, the photon is its own antiparticle so that .

Definition: By 13.8 , the photon field is

15.1

which is hermitian, so only one photon field is necessary in the current, so is permissible and photons

can be absorbed and emitted singly. The commutator is

15.2

Thus, by 12.10 and 13.4

15.3

The constraint that has only components of spin α is necessary if the interaction operator creates

eigenstates of spin. This is observed; we assume that it also holds for time-like and longitudinal spin. Thentransforms as (defined in 11.1 ) under space inversion. So

15.4

since has no space-like component and for r = 1,2,3 has no time like component. Now

substitute p →- p in the second term of 15.3 at x0 = y0

15.5Then by substituting in 14.2 , and noting from 13.7 that the commutation relationship with the inter-

action density is determined by the commutation relationship with the current

15.6

The physical interpretation of 15.6 is that observable effects associated with photons depend only on

changes in photon number; since photons can be absorbed or emitted singly the number of photons cannot

be an eigenstate of an operator constructed from the interaction and cannot therefore be known. Let φµ( x)be a gauge term, that is an arbitrary solution of

15.7

having no physical meaning. Then physical predictions from 15.6 are invariant under the gauge transforma-

tion , and the value of is hidden by the gauge term. Differentiating 15.6 using14.2 gives

15.8

Differentiate twice and observe that for the photon so . Then from 15.1

15.9

x α,| x α,| =

Aα x( ) x α,| x α, |+=

J A=

Aα x( ) Aβ y( ),[ ] x α,| x α, |+ y β,| y β, |+,[ ]= x α, y β, | = y β, x α, | –

Aα x( ) Aβ y( ),[ ] η r ( ) d 3 pΜ∫

r ∑ α p r , | e i– p x y–( )⋅ p r , β | β p r , | e ip x y–( )⋅ p r , α | –=

Aα x( )

α p r , | wα p r ,( )β p– r , | p– r , α | αp r , | p r , β | =

wα p 0,( ) wα p r ,( )

A x( ) A y( ),[ ] x0 y0= 0=O A=

Aβ x( ) α∂ Aα β∂ x( ) =

φµ µ

x( )∂ 0=

A x( ) A x( ) φ x( )+→ A x( )

A x( ) 2∂ Aα∂ x( ) α∂ i H x( ) A0∂ x( ),[ ] A2∂ x( ) += =

p 2 0= x α,| 2∂ 0=

A2∂ x( ) 0=




Then 15.8 reduces to

15.10

Given H , 15.10 can be calculated from the commutator between the fields

15.11

But by 13.5 and 13.4

15.12

and

15.13

Substituting p →- p in 15.13 at x0 = y0 and using 15.4 and 15.11 gives, for the space-like components of the

derivative

For ,

and for the time-like component

15.14

Theorem: The equal time commutator 15.14 satisfies locality, 13.16 , if

15.15

Proof: It follows from 15.15 that

15.16

Substituting 15.16 into 15.14 shows locality is satisfied by the equal time commutator

15.17

Substituting 15.15 into 15.1 using 12.3 gives the photon field

15.18

By 15.16 , 13.2 and 12.11

15.19

So the commutator, 15.2 , is

15.20

A x( ) 2∂ i H x( ) A0∂ x( ),[ ] =

Aα x( ) Aβ y( ),∂[ ] x α,∂ y β, | y β, x α,∂ | –=

x α,∂ y β, | ηr ( ) d 3 p

Μ∫

r 0=

3

∑– α p r , | p r , β | ipe i– p x y–( )⋅=

y β, x α,∂ | ηr ( ) d 3 p

Μ∫

r 0=

3

∑ β p r , | p r , α | ipe ip x y–( )⋅=

i 1 2 3, ,= Ai x( ) A y( ),∂[ ] x0 y0= 0=

A0 α x( ) Aβ y( ),∂[ ] x0 y0= 2 i η r ( ) d 3 p

Μ∫

r 0=

3

∑– α p r , | p r , β | p 0 e i p x y–( )⋅=

α p r , | 12π------ 3

2--- wα p r ,( )

2 p 0

--------------------=

η r ( ) α p r , | p r , β | r 0=

3

∑ η r ( )δαβ16 π3 p 0--------------------=

A0 x( ) A y( ),∂[ ] x0 y0= ig δ xy–=

Aα x( ) η r ( ) d 3 p

2 p 0

------------Μ∫ r 0=

3

∑e ip x⋅ p r ,| e i– p x⋅ p r , |+( )wα p r ,( )=

x α, y β, | gαβ8π3--------- d 3 p

2 p 0-------- e i– p x y–( )⋅∫ =

Aα x( ) Aβ y( ),[ ] gαβ8π3--------- d 3 p

2 p 0--------

Μ∫ e i– p x y–( )⋅ e ip x y–( )⋅–( )=




Theorem: 15.20 is zero outside the light cone.Proof: The proof follows the text books, e.g. [1], and is left to the reader.Theorem: satisfies the Lorentz gauge condition

15.21

Proof: by 15.6

by differentiating 15.18 . But this is zero which establishes 15.21 .

16 The Dirac Field

Definition: By 13.8 , the Dirac field is16.1

We know from observation that a Dirac particle can be an eigenstate of position. Any physical configurationcan only be a combination of particle interactions so it is possible to form the position operator

16.2

from the current 13.10 , for any region X which can be as small as the apparatus will allow. Position kets area basis, so 16.2 reduces to

up to the resolution of the apparatus. Current can only generate eigenstates of spin and position if it does notmix basis states, so

16.3

Then by 12.1

16.4

and by 18.8

16.5

Definition: The Dirac adjoint of the annihilation operator is

16.6

A x( ) Aα x( ) α∂ 0=

Aα x( ) α∂ Aα α∂ x( ) =

η r ( ) d 3 p

2 p 0

------------Μ∫

r 0=

3

∑ e ip x⋅ p r ,| e i– p x⋅ p r , |+( )i p α p α–( )wα p r ,( ) =

ψ α x( ) x α,| x α, |+=

Z X ( ) x| x | x X ∈∑=

Z x( ) x| x |=

x N ∈∀ x α,| x α,| =

α p r , | 12π------

32---uα p r ,( )=

x α, |12π------

32---

d 3 p

Μ∫ r ∑ uα p r ,( )ei– p x⋅

p r , |=

x α, |

x α,| x µ,| γ µα0

µ∑ 1

2π------

32---

d 3 p

Μ∫

r ∑ uα p r ,( )e ip x⋅ p r ,| = =




Similarly by 12.2

16.7

and by 18.8

16.8

Definition: The Dirac adjoint of the creation operator is

16.9

Definition: The Dirac adjoint of the field is

16.10

Theorem: The anticommutation relations for the Dirac field and Dirac adjoint and obey

16.11

16.12

Proof: 16.11 follows from the definitions, 16.1 and 16.10 By 6.10 and 5.16 we have

16.13

where T denotes that α and β are transposed. By 16.5 and 16.6 , and using 12.11 .

16.14

by 10.7 . Likewise for the antiparticle, by 16.8 and 16.9

16.15

by 10.12 . Substituting p →- p at x0 = y0 in 16.15 gives

16.16

So, by 16.13 , adding 16.14 and 16.16 at x0 = y0 gives the equal time anticommutator

16.17

α p r , | 12π------

32---vα p r ,( )=

x α,| 12 π------ 3

2---

d 3 p

Μ∫ r ∑ vα p r ,( )e ip x⋅ p r ,| =

x α,|

x α, | x µ, |γ µα0

µ∑ 1

2π------

32---

d 3 p

Μ∫

r ∑ vα p r ,( )e ip x⋅ p r ,| = =

ψ ˆ α x( ) ψ †µ x( )γ µα0 x α,| x α, |+= =

ψ ν x( ) ψ λ y( ),{ } ψ ˆ µ x( ) ψ ˆ κ y( ),{ } 0= =

ψ α x( ) ψ ˆ β y( ),{ } x0 y0= γ αβ0 δ xy=

ψ α x( ) ψ ˆ β y( ),{ } x α, | y β,| ,{ } x α,| y β, |,{ }+=

x α, y β, | y β, x α, | T+=

x α, y β, | 18π3--------- d 3 p

Μ∫

r ∑ uα p r ,( )uβ p r ,( )e i– p x y–( )⋅=

18π3--------- d 3 p

2 p 0--------

Μ∫ = p γ ⋅ m+( )αβe i– p x y–( )⋅

y β, x α, | Τ 18π3--------- d 3 p vα p r ,( )vβ p r ,( ) e ip y ix p⋅–⋅

Μ∫

r ∑=

1

8π3---------= d 3 p

2 p 0-------- p γ ⋅ m–( )

αβ e ip x y–( )⋅

Μ∫

y β, x α, | x0 y0=1

8π3--------- d 3 p

2 p 0-------- 2 p 0γ 0 p γ m–⋅–( )e i– p x y–( )⋅

Μ∫ =

ψ α x( ) ψ ˆ β y( ),{ } x0 y0=1

8π3--------- γ αβ

0 d 3 p e i– p x y–( )⋅

Μ∫ =




16.12 follows immediately.Theorem: The anticommutation relations for the Dirac field and the Dirac adjoint obeys locality, 13.16 .

Proof: By 16.14

16.18

And by 16.15

16.19

By 16.13 the anticommutator is found by adding 16.18 and 16.19

16.20

16.20 is and zero outside the light cone. The proof follows the text books, e.g. [1], and is left to the reader.

17 The Non-Perturbative SolutionBecause local phase is a freedom in the definition of Hilbert space and can give no physical results we

have that U (1) local gauge symmetry is preserved in interaction ( section 2 ). Then we have the intuitivelyappealing minimal interaction characterised by the emission or absorption of a photon by a Dirac particle.

According to 13.7 an interaction H between photons and Dirac particles is described by a combination of

particle currents, which, by 13.10 , are themselves hermitian combinations of particle fields.Definition: The photon current operator is A(x)

Definition: The Dirac current operator is

17.1

Postulate: Let e be the electromagnetic coupling constant. The electromagnetic interaction density is

17.2

Lemma:

17.3

Proof: Using the definitions 16.1 and 16.10 to expand 17.1

17.4

where the summation convention is used for the repeated indices, µ and ν. In classical situations we onlyconsider states of a definite number of Dirac particles, so the expectation of the pair creation and annihila-tion terms is zero by 4.1 . Using 16.5 and 16.6 and differentiating the particle term in 17.4

x α, y β, | 18π3--------- i ∂ γ ⋅ m+( ) d 3 p

2 p 0-------- e i– p x y–( )⋅

Μ

∫ =

y β, x α, | T 18π3---------– i ∂ γ ⋅ m+( ) d 3 p

2 p 0-------- e ip x y–( )⋅

Μ∫ =

ψ α x( ) ψ ˆ β y( ),{ } 18π3--------- i ∂ γ ⋅ m+( ) d 3 p

2 p 0--------

Μ∫ e i– p x y–( )⋅ e ip x y–( )⋅–( )=

jα x( ) :ψ ˆ µ x( )γ µνα ψ ν x( ): :ψ ˆ x( )γ αψ x( ):= =

H x( ) ej x( ) A x( )⋅ e :ψ ˆ x( )γ A x( )ψ x( ):⋅= =

∂ j⋅ x( ) 0=

jα x( ) x µ,| γ µνα x ν,| x µ,| γ µν

α x ν, | γ µνα x ν,| x µ, |– x µ, |γ µν

α x ν, |+ +=

x µ,| γ µνα x ν, |α∂ 1

8π3--------- d 3 p d 3 q iu p r ,( ) q γ ⋅ p γ ⋅–( )u q s,( )e ix q p–( )⋅ p r ,| q s, |

Μ∫

Μ∫

r s,∑=




Using 16.8 and 16.9 and differentiating the antiparticle term in 17.4

Here v and have been ordered so that the spin index can be unambiguously omitted. 17.3 follows by dif-

ferentiating 17.4 and using 10.6 and 10.11 .Lemma:

17.5

Proof:

17.6

by 16.12 . Take the hermitian conjugate and apply 10.5

Postmultiply by γ 0

17.7

So, by commuting the terms,

using 17.6 and 17.7 . 17.5 follows from 10.5Theorem: is a classical conserved current, i.e.

17.8Proof: Substituting in 14.3

17.8 follows from 17.3 and 17.5 , so is conserved.Theorem: can be identified with classical electric charge density, i.e.

, 17.9

Proof: It is straightforward from 6.2 that j is additive for multiparticle states, so it is sufficient to show the

theorem for a one particle state . By 17.4

by ordering terms so that the spinor indices can be suppressed. Then 16.14 follows from 16.3 and 10.5

Except in so far as #ë was used to justify an analysis of measurement, classical law does not form part

of the assumptions, and according to .æ , the claim that the minimal interaction is the cause of the electro-magnetic force requires:

γ µνα x ν,| x µ, |α∂ 1

8π3--------- d 3 p d 3 q iv q r ,( ) p γ ⋅ q γ ⋅–( )v p s,( )e ix p q–( )⋅ p r ,| q s, |

Μ∫

Μ∫

r s,∑=

v

j0 x( ) jα x( ),[ ] 0=

ψ x( ) j, α x( )[ ] ψ x( ) :, ψ ˆ x( )γ αψ x( ):[ ]= ψ x( ) ψ ˆ x( ),{ }γ αψ x( )=

γ 0γ αψ x( )=

jα x( ) ψ †, x( )[ ] ψ † x( )γ α†γ 0 ψ ˆ x( )γ α= =

jα x( ) ψ ˆ, x( )[ ] ψ ˆ µ x( )γ αγ 0=

j0 x( ) jα x( ),[ ] :ψ ˆ x( )γ 0ψ x( ): jα x( ),[ ]=

ψ ˆ x( )γ 0 ψ x( ) j, α x( )[ ] ψ ˆ x( ) j, α x( )[ ]γ 0ψ x( )+=

ψ ˆ x( )γ 0γ 0γ αψ x( ) ψ ˆ x( )γ αγ 0γ 0ψ x( )–=

j

j x( ) ⋅∂ 0=O j α=

jα x( ) a∂ i H x( ) j0 x( ),[ ] jα α∂ x( ) +=

j j0

f | ) ∈∀ j0 x( ) x f | 2 f x | 2–=

f | *∈

j0 x( ) f x µ, | γ µν0 x ν, f | γ µν0 f x ν, | x µ, f | –=

f x | γ 0γ 0 x f | x f | γ 0γ 0 f x | –=




Theorem: satisfies Maxwell’s Equations

17.10

Corollary: Maxwell’s equations simplify immediately to their form in the Lorentz gauge

17.11

Proof: By 15.21 it is sufficient to prove the corollary. By 15.10 and 17.2

17.11 follows immediately from 15.17 .Theorem: (Classical gauge invariance). Let g be an arbitrary differentiable function. Then classical prop-

erties are invariant under gauge transformation of the photon field given by

17.12

Proof: It is a well known result following from 17.10 that the classical properties of the electromagnetic

field depend only on derivatives of , defined by

Then is clearly invariant under 17.12 . Although classical electrodynamics is gauge invariant, the

Lorentz gauge, 15.21 , is here theoretically determined and we have .

18 Feynman Rules

Definition: For any vector p, such that , let be a matrix for any . satisfies

the identity

18.1

Lemma: For , we have the identities

18.2

18.3

and for we have the identities

18.4

18.5

Proof: These are evaluated as contour integrals and the proofs are left to the reader.

A x( )

Aα x( ) Aµ x( ) µ∂α∂–2∂ e jα x( ) –=

A x( ) 2∂ e j x( ) –=

A x( ) 2∂ i j x( ) A x( )⋅ A0∂ x( ),[ ] =

Aα x( ) Aα x( ) gα x( )∂+ → Aα x( ) gα x( )∂+=

A x( ) F αβ Aβ x( ) α Aα x( ) β∂–∂≡

F αβgα∂ 0=

p 2 m2= p p 0 p,( )= p 04 ∈ p

p 02 p 0

2– p 2 m2–≡ x 0> ε 0>

e i p 0 iε–( ) x

2 p 0 iε–( )------------------------ i–2π------ dp 0

∞–

∞∫ e ip 0 x–

p 02 p 0 iε–( )2–

----------------------------------- i–2π------ dp 0

∞–

∞∫ e i p0 x–

p 2 m2– 2 ip 0ε ε2+ +------------------------------------------------- -≡ ≡

e i p0 iε–( ) x

2---------------------- i–

2π------ dp 0∞–

∞∫

p 0 e i p 0 x–

p 2 m2– 2ip 0ε ε2+ +------------------------------------------------- -≡

x 0< ε 0>

e i p0 iε–( ) x–

2 p 0 iε–( )------------------------ i–2π------ dp 0

∞–∞∫ e ip

˜0 x–

p 2 m2– 2ip 0ε ε2+ +------------------------------------------------- -≡

e i– p 0 iε–( ) x

2------------------------ i–

2π------ dp 0∞–

∞∫

p 0 e i p0 x–

p 2 m2– 2ip 0ε ε2+ +------------------------------------------------- -≡






and for an antiparticle, by 16.7

18.8

Similarly for final particles in the state connected to the node gives for a photon

for a Dirac particle

and for an antiparticle

18.9

The time ordered product in 13.15 leads to an expression for the photon propagator

18.10

18.10 is to be compared with the decomposition of distributions into advanced and retarded parts according

to the method of Epstein and Glaser which also excludes [17][5] . Indeed our analysis of the origin

of the ultraviolet divergence is essentially the same as that given by Scharf [17] . The difference between this

treatment and Scharf is that our limiting procedure uses a discrete space and here the “switching off and

switching on” of the interaction at is a physical constraint meaning that only one interaction takes

place for each particle in any instant, as discussed in section 7 . By 15.19 18.10 is

Use 18.2 in the first term, recalling that , and use 18.4 and substitute in the second term.

Then we have

18.11

For each node the Dirac current generates two propagators, one for the field and one for the adjoint. The

field either annihilates a particle or creates an antiparticle, and is represented by an arrowed line pointing

towards the vertex. The field at vertex n acting on vertex j, generates the propagator arrowed from

j to n

18.12

The Dirac adjoint field creates a particle or annihilates an antiparticle, and is represented by an arrowed line

pointing away from the vertex. The adjoint generates the propagator arrowed from n to j

18.13

xn α, p r , | 12π------ 3

2--- vα p r ,( )e i– p x n⋅=

p r , | xn

12π------ 3

2--- wα p r ,( )

2 p 0

-------------------- e ip x n⋅

12π------ 3

2--- uα p r ,( )e ip x n⋅

12π------ 3

2--- vα p r ,( )e ip x n⋅

Θ x0n x0

j–( ) xn α, x j β, | Θ x0 j x0

n–( ) x j β, xn α, | T+

x0n x0

j=

x0n x0

j=

gαβ8π3--------- d

3 p2 p 0

-------- Θ x0n x0 j–( )e i– p xn

x j

–( )⋅ Θ x0 j x0n–( )e ip xn

x j

–( )⋅+[ ]Μ∫

m2 0= p–→

i–gαβ

16 π4------------ d 3 p

2 p 0-------- Lim

ε 0 +→ dp 0∞–

∞∫ Θ x0

n x0 j–( ) Θ x0

j x0n–( )+[ ] e i– p xn x j–( )⋅

p 2 2ip 0ε ε2+ +-------------------------------------

Μ∫

ψ α xn( )

Θ x0n x0

j–( ) xn α, x j β, | Θ x0 j x0

n–( ) x j β, xn α, | T–

ψ ˆ α xn( )

Θ x0n x0

j–( ) xn α, x j β, | Θ x0 j x0

n–( ) x j β, xn α, | T–




The time ordered product in 13.15 is unaffected under the interchange of and . By inter-

changing and in the diagram, we find for the adjoint propagator arrowed from j to n

18.14

18.14 is identical to 18.12 , the expression for the Dirac propagator arrowed from j to n, so we do not distin-

guish whether an arrowed line in a diagram is generated by the field or the adjoint field. Similarly we findthat the photon propagator, 18.10 is unchanged under interchange of the nodes, so we identify all diagrams

which are the same apart from the ordering of the vertices and remove the overall factor for a diagramwith n vertices. By 16.14 and 16.15, 18.12 is

18.15

Use 18.2 and 18.3 in the first term, and use 18.4 and 18.5 and substitute in the second term. Then

the propagator 18.15 is

Now collect all the exponential terms with in the exponent under the sum 18.6, and observe that the sum

over space is a momentum conserving delta function by 3.12. Then integrate over momentum space and

impose conservation of momentum at each vertex, leaving for each independent internal loop

18.16

Only the time component remains in the exponents for the external lines 18.7 - 18.9. Introduce a finite cutoff

by writing the improper integral

and instructing that the limits , should be taken after calculation of all formulae. Then the

photon propagator, 18.11 reduces to

18.17

For a Dirac particle, , so we can also simplify the denominator by shifting the pole under the limit

. Thus the Dirac propagator arrowed from j to n is

18.18

The propagators, 18.17 and 18.18 , vanish for , and are finite in the limit , since the inte-

grands oscillate and tend to zero as . Loop integrals ( 18.16 ) are proper and the denominators do not

vanish so the ultraviolet divergence and the infrared catastrophe are absent, provided that the limitsand are not taken prematurely. In the denominator of 18.17 , ε2 plays the role of the small photon

xn α,( ) x j β,( ) xn α,( ) x j β,( )

Θ x0 j x0

n–( ) x j β, xn α, | T Θ x0n x0

j–( ) xn α, x j β, | +

1 n! ⁄

Θ x0n x0

j–( )8π3

-------------------------- d 3 p

2 p 0--------

Μ∫ ip γ ⋅ m+( )e i– p x n x j–( )⋅

Θ x0 j x0

n–( )8π3

-------------------------- d 3 p

2 p 0--------

Μ∫ ip γ m–⋅( )e ip x n x j–( )⋅+

p–→

i– gαβ16 π4------------ d

3 p

2 p 0-------- Limε 0 +→ dp 0

∞–

∞

∫ Θ x0n x0 j–( ) Θ x0 j x0n–( )+[ ] ip γ ⋅ m+( )ei– p xn x j–( )⋅

p 2 m2– 2ip 0ε ε2+ +------------------------------------------------------ -

Μ∫ x n

18π3--------- d 3 p

2 p 0--------

Μ∫

Λ 0∈

i–2π------ dp 0

∞–

∞∫ Lim

Λ ∞→ dp 0Λπ–

Λπ∫ =

Λ ∞→ ε 0 +→

ig αβ2π----------– dp 0

Λπ–

Λπ∫

1 δ x0n x0

j–( )e ip 0 x0n x0

j–( )

p 2 2ip 0ε ε2+ +--------------------------------------------------

p 0 0>ε 0+→

i–2π------ dp 0

Λπ–

Λπ∫

1 δ x0n x0

j–( ) p γ ⋅ m+( )αβe i x0n x0

j–( ) p 0–

p 2 m2– iε+------------------------------------------------------------------------------------

x0 j x0

n= Λ ∞→ p 0 ∞→

Λ ∞→ε 0+→




mass commonly used to treat the infrared catastrophe, and as with Scharf’s treatment there is no additional

requirement to include a photon mass. The standard rules are obtained by ignoring in the numerator

of 18.17 and 18.18 , and observing that for , the sums over time in the vertex 18.6 act as conserv-

ing δ functions ( is not energy since energy was defined in section 8 to be on mass shell, but is equal

to the energy of the measured, initial and final, states).

Thus the discrete theory modifies the standard rules for the propagators and justifies the subtraction of divergent quantities, but here this is no ad hoc procedure but regularisation by the subtraction of a term

which recognises that a particle cannot be annihilated at the instant of its creation. To see that this subtrac-tion is effectively the same as that described by Scharf we replace in the propagators 18.10 and 18.12

with a monotonous function over with

Then we observe that in the limit as the discrete unit of time the sums over space become integrals

since these are well defined [17] . In the limit Μ is replaced by and 18.10 and 18.12 are replaced bydistributions which have been split with causal support. When the distributions are combined in 18.17 and

18.18 we obtain the usual Feynman rules, together with a term coming from , which gives a distribu-tion with point support in the limit (c.f. Scharf [17] 3.2.46). The most straightforward way to determine the

effect of this term is to consider the non-perturbative solution ( section 17 ). This allows us to impose threeregularisation conditions on the propagator, that it is independent of lattice spacing χ at low energies and

the renormalised mass and charge adopt their bare values, since bare mass and charge appear in Maxwell’s

equations.

This in no way contradicts the calculation that the apparent or “running” coupling constant exhibits an

energy dependency in scattering due to perturbative corrections. But it shows that this dependency is

removed in the calculation of the expectation of the current, and enables us to regularise the theory to thelow energy value. The calculation of effective charge [15] .(7.96) by the summation of one particle irreduc-ible insertions [15] .(7.94) into the photon propagator breaks down to any finite order, so the limit may not

be taken. More generally the renormalisation group arguments leading to the behaviour of the Callan-

Symanzik equation depend upon the sum of a geometric series [15] .(10.27) which does not converge at theLandau pole. The Landau pole is absent in a model in which there is a fundamental minimum unit of time

since high energies correspond to short interaction times and in a discrete model this implies fewer interac-tions. In the limit as the discrete unit of time goes to zero the pole suggests nothing more serious than the

failure of an iterative solution, not the failure of the model.

19 Yang-Mills Fields and Quark ConfinementFollowing the arguments of section 9 we seek to replace the discrete time evolution equation 7.2 with a

covariant equation of motion, which should have the form of 9.2 , but with the addition of an interaction term

19.1

Now we seek to generalise to the case where we do not have a simple Dirac particle, but a composition of

Dirac particles where i is an index (colour), and is a flavour of quark. There are two straight-

δ x0n x0

j

Λ 0∈ p 0

p 0 p 0

ΘC ∞ χ0

4 1

χ0 t ( ) 0 for t 0≤1 for t χ≥

=

χ 0→4 3

δ x0n x0

j

i Γ H x( )+⋅∂( ) f m f =

x i q iµi,| q i




forward ways of forming states which are symmetrical in the colour index. The usual version of SU(3) is

formed by creating linear combinations of quark states , but it is interesting also to consider a model

built from multiplets of quarks. Such models are generally rejected as being non-renormalisable, but deeperanalysis will show that this is not so because the quarks are decoupled within the multiplet, and that the

model has similar properties to the standard model.

Whether quarks are confined by the physical interactions described here, or whether there is some deeperconfining mechanism in qcd, the current treatment provides a model with the principle features of the stronginteractions, and is applicable whenever substructure is not relevant. A general state of three quarks in * ,

as defined in 4.3

19.2

has a co-ordinate for the position of each quark. But the physical space is the subspace of * generated

from 19.1 . So physical states are either directly created from the creation operators appearing in , or

evolve from them from the corresponding non-interacting a wave equation.

The spin statistics theorem forbids even multiplets but it is possible to build a model in which the fieldoperators create or annihilate triplets of quarks. There is no way physically to distinguish the three quarks

in a hadron, so the creation operator must be (anti)symmetric under permutations of the three quarks. The

physical space is generated by creation operators of the form

19.3

where the semi-colons indicates that the state has been symmetrised. We write the creation operator morecompactly

Then the field operator for a baryon is

19.4

This expression for the field operator is formally like that used when we consider colour compositions of singlet quarks, and to that treated in the literature, e.g. [15] (15.19). We draw attention to this because,

although formally similar, the symmetry states are interpreted a little differently from the standard model

where fields are considered separately from the Fock space on which they act. Historically SU (2), or isospin,

was introduced by considering a nucleon as a particle which could be either proton or neutron. But the sym-

metry in 19.4 was found by considering a particle which is a composition of colours. In other words 19.4

represents the field operator for a particle containing three colours in symmetrical combinations, not one of

any colour with symmetrical probability amplitude.Following the construction of qed we seek to construct an interaction operators in which a vector

boson is emitted or absorbed by the baryon. The treatment of the electromagnetic interaction follows thatfor leptons, but the Dirac adjoint is

and there is a vector current for each quark

xi q iµi,|

f | x1 q1 µ, 1,| x2 q2 µ2, ,| x3 q3 µ3, ,| ,,( )=

xi

H x( )

f | x q 1 µ, 1 x q2 µ2 x q3 µ3, ,;, ,;,| =

f | x i q i µ, i, ,| x q1 µ, 1 q2 µ2 q3 µ3,;,;,| = =

ψ x( ) ψ iµ x( ) x i q i µ, i, ,| x i q i µ, i, , |+= =

H x( )

ψ ˆiµ x( ) ψ †

iα x( )γ αµ0=

jαi x( ) :ψ ˆ

iµ x( )γ µνα ψ i ν x( ):=




Then we can write down the electromagnetic interaction density

19.5

where is the charge of and we sum over index so the interaction is colourless. To see that 19.5

reduces to the standard treatment of the electromagnetism we need to establish that the quarks of each colourare decoupled in interactions.

Theorem: In the electromagnetic interaction, in the case when the hadron is preserved (i.e. ignoring pair

creation and pair annihilation) the photon interacts with one quark, and leaves spin and momentum of theother quark(s) unchanged.

Proof: It is sufficient to demonstrate the theorem for . The proof for and 3 is identical. The

term of 19.5 is

19.6

Then suppressing in 19.4 , 19.6 can be written

19.7

Ignoring pair creation and annihilation and the antiparticle term, the interaction of a particle is

which is in matrix notation

+ colour symmetry states 19.8

Using the resolution of unity and 16.3 , which holds for Dirac particles, 19.8 shows that the first quark par-

ticipates in an interaction with an identical form to the electromagnetic interaction for leptons, while theother two particles are unaffected.

It has been observed that local U (1) transformations have no physical meaning in the description of

Hilbert space, so that this must be a symmetry group. Under the U (1) transformation

19.9

19.1 transforms to

19.10

since commutes with which is a function, not an operator. Then we can restore 19.1 by

Then replacing A by its expectation and using 17.12 shows that 19.1 is invariant for the classical electro-magnetic field.

H x( ) e i j i x( ) A x( )⋅=

e i q i i

i 1= i 2=i 1=

e i :ψ ˆiµ x( )γ µν

α ψ i ν x( ): Aα x( ) x Ν∈∑q i

:e1 x µ1 µ2 µ3;;,| x µ1 µ2 µ3;;, |+( )γ µ1 ν A x( )⋅ x ν µ2 µ3;;,| x ν µ2 µ3;;, |+( ): x Ν∈∑

:e1 x µ1 µ2 µ3;;,| γ µ1 ν A x( )⋅ x ν µ2 µ3;;, |: x Ν∈∑

e1 x µ1,| γ µ1 ν′γ ν′ν0 A x( )⋅ x ν, |

x Ν∈∑

x µ2,| x µ2, | x Ν∈∑ x µ3,| x µ3, | x Ν∈∑

f e iα x( ) f →

e iα x( ) i Γ Γα x( )⋅∂– H x( )+⋅∂( ) f me iα x( ) f =

H x( ) e iα x( )

A x( ) A x( ) Γα x( )⋅∂+→




The electromagnetic interaction, 19.5 , preserves colour symmetry because there is an interaction term in

each colour index. This is the subgroup. There is also an subgroup for states consisting of

three quarks. So we may also try a Yang-Mills interaction in the form of 19.5 , but with a current defined foreach of the eight generators, , of

19.11

Because these currents mix flavours the same coupling constant is needed for all quarks, and the interaction

density takes the form

where are the gluon fields, which are identical to the photon, up to interactions. Then the equation of motion, 19.1 is

19.12

As with qed the equation of motion, 19.12 , can be read as an eigenvalue equation for the gauge covariantderivative showing the origin of gauge invariance. The pure boson vertices are absent from 19.12 . Bosonic

interactions violate the Ward identity which follows immediately from the equation of motion 9.7 for inter-

mediate vector bosons. Normally the Faddev-Popov ghosts restore the Ward identity but in this model the

pure boson vertices are also understood as ghost interactions arising from the quantisation of a gauge field

which has no physical meaning.

In the absence of interaction the equation of motion, 19.1 , shows that states in the form of 19.2 evolve as

three separate Dirac particles; since quarks may have different masses the wave functions cannot in generalremain identical. Nonetheless quarks are confined in measured states because there is only one space co-

ordinate in the interaction density for all the quarks in an interaction. If one quark is localised, for exampleby an electromagnetic interaction with other matter such as might be used in a measurement of position,

then the other quarks are confined at the same point when the interaction takes place. Confinement of thissort affects measured states, and should not be seen as a new law of quantum mechanics but rather as aninstance of the collapse of the wave function. The interactions which go towards measurement are the same

as other interactions, and the projection operator corresponding to the collapse of a wave function is always

a composition of currents such as 19.11 .

By following through the derivation of Feynman rules ( section 18 ) we observe that the interaction gen-erates a momentum conserving delta function for each individual quark. Thus for hadrons in which the

quarks are in eigenstates of momentum, the interaction leaves the momenta and the spin states of two quarks

in a baryon unchanged, as though only one of the three quarks participates in the interaction. Thus quarksare `quasi free’ – they are confined in co-ordinate space, and yet have values of momentum independent of

each other.

AcknowledgementsI should like to thank a number of physicists who have discussed the content and ideas of this paper on

usenet, particularly Paul Colby, Matthew Nobes, Michael Weiss and Toddlius Desiato for their constructive

criticism of earlier versions of the paper, John Baez for instruction and advice about the current status of

field theory, and the moderators of sci.physics.research (John Baez, Matt McIrvin, Ted Bunn & Philip Hel-

big) for their vigilance in pointing out lack of clarity in expression in describing the model.

U 1( ) SU 3( )

t a SU 3( ) j

αa x( ) :ψ ˆ

iµ x( )t

ij

a γ µνα ψ

j ν x( ):=

H x( ) gj a x( ) Aa x( )⋅=

A a x( )

i γ gj a x( ) A a x( )⋅+⋅∂( ) f m f =




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物理学经典书籍 Discrete Quantum Electrodynamics - C.Francis

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